PCM20210811_SICP_1.3.1.1_Procedures_as_Arguments_I.jl

### A Pluto.jl notebook ###
# v0.19.27

using Markdown
using InteractiveUtils

# ╔═╡ f1c1495e-4203-4d73-a96b-f47e3e54d865
using Plots, QuadGK, LaTeXStrings, Statistics

# ╔═╡ 015f86b0-fa99-11eb-0a32-5bee9e7368fb
md"
=====================================================================================
#### SICP: [1.3.1.1 Procedures as Arguments I](https://sarabander.github.io/sicp/html/1_002e3.xhtml#g_t1_002e3_002e1): Basics (e.g. Cantor Set, Integration à la Riemann & Lebesgue (NonSICP))
##### file: PCM20210811\_SICP\_1.3.1.1\_Procedures\_as\_Arguments\_I.jl

##### Julia/Pluto.jl-code (1.9.3/19.27) by PCM *** 2023/11/12 ***
=====================================================================================
"

# ╔═╡ acd01166-5258-4012-a6f4-e4f05b49dce3
md"
##### 1.3.1.1 SICP-Scheme-like *functional* Julia
"

# ╔═╡ e3414514-9956-416e-ad41-503795d32556
md"""
---
$cube: (\mathbb N \cup \mathbb R) \times (\mathbb N \cup \mathbb R) \rightarrow (\mathbb N \cup \mathbb R)$
$\;$

$x \mapsto cube(x)$

$\;$

$cube(x) := x^3$

$\;$

"""

# ╔═╡ f63fda47-0ee7-4fac-b90c-b4bd893daa7d
cube(x) = *(x, x, x)              # '*' is prefix operator   (SICP, p.57)

# ╔═╡ 7fcc1606-a914-458a-82ab-a3fa4f1b0b49
cube(3)                           # result is Integer

# ╔═╡ 590b9e77-9983-4e54-9914-4d6bb5ce8336
cube(3.)                          # result is Float

# ╔═╡ 6414c834-cfe4-4208-a4b1-7fbf3ced47df
md"
---
$sumIntegers : (\mathbb N \cup \mathbb R) \times (\mathbb N \cup \mathbb R) \rightarrow (\mathbb N \cup \mathbb R)$

$\;$

$(a, b) \mapsto sumIntegers(a, b)$

$\;$

$sumIntegers(a, b) := \sum_{i=a}^{b}i$

$\;$
$\;$
$\;$

"

# ╔═╡ df6d6fdc-20f1-4ead-8e3e-07340ddb521a
sumIntegers1(a, b) =                         # (SICP, p.57)
	>(a, b) ? 
		0 : 
		+(a, sumIntegers1(+(a, 1), b))

# ╔═╡ 5e22d368-1433-4ea1-b094-0188cb755c89
sumIntegers1(0, 0)                           # result is Integer

# ╔═╡ 27d6dbc2-349d-4848-a915-031f42b72d91
sumIntegers1(0., 0)                          # result is Float

# ╔═╡ 99bc5602-46d7-401e-a625-2efa10806042
sumIntegers1(0, 0.)                          # result is Integer

# ╔═╡ 3a7dc73f-9435-40ec-a58c-09600dcb834d
sumIntegers1(0., 0.)                         # result is Float

# ╔═╡ 0b466d51-d50d-45bf-8290-31414a36c3c5
sumIntegers1(2, 5)                           # result is Integer

# ╔═╡ 4e086e86-51c8-47ec-8810-55c253f3c92c
sumIntegers1(2., 5)                          # result is Float

# ╔═╡ dbaa23c2-accd-478e-83da-cb9383a2557b
sumIntegers1(2, 5.)                          # result is Integer

# ╔═╡ 381b81f0-7f5f-4312-b63a-fb654e1668e6
sumIntegers1(2., 5.)                         # result is Float

# ╔═╡ 0f94752d-afc7-4c19-96b8-f0e40edd8f6f
md"
---
$sumCubes : (\mathbb N \cup \mathbb R) \times (\mathbb N \cup \mathbb R) \rightarrow (\mathbb N \cup \mathbb R)$
$\;$
$(a, b) \mapsto sumCubes(a, b)$
$\;$
$sumCubes(a, b) := \sum_{i=a}^{b}i^3$
$\;$
$\;$
$\;$
"

# ╔═╡ a92d3b32-fc1c-4bd9-ba16-d650ffc805eb
sumCubes1(a, b) =                            # (SICP, p.57)
	>(a, b) ? 
		0 : 
		+(cube(a), sumCubes1(+(a, 1), b))

# ╔═╡ a8fbd1f3-9994-4e68-83e1-5272aa85f52f
sumCubes1(0, 1)                              # result is Integer

# ╔═╡ ce73399c-aa70-4116-beb2-4f861fc01031
sumCubes1(0, 3)                              #  0+1+2*2*2 + 3*3*3 = 1 + 8 + 27 = 36

# ╔═╡ a890a15b-e0ac-4336-b666-17389f7bae8a
sumCubes1(1, 4)                              #  36 + 4*4*4 = 36 + 64 = 100

# ╔═╡ 43ca7083-657d-499e-b273-0da40720342a
sumCubes1(1., 4)                             # result is Float

# ╔═╡ fe6b263c-a804-4e64-9f19-78fe146abc5b
sumCubes1(1, 4.)                             # result is Integer

# ╔═╡ f33b6724-4bad-4b46-aff7-2763f51fb1a8
sumCubes1(1., 4.)                            # result is Float

# ╔═╡ 0c1a55e9-5c9f-4613-bc04-8a9959450a9b
md"
---
$\pi_{Sum}: (\mathbb N \cup \mathbb R) \times (\mathbb N \cup \mathbb R) \rightarrow (\mathbb N \cup \mathbb R)$
$\;$
$\pi_{Sum}: (a, b) \mapsto \pi_{Sum}(a, b)$

$\;$

$\pi_{Sum}(a=1, b) := \frac{1}{1\cdot3}+\frac{1}{5\cdot7}+\frac{1}{9\cdot11}+... \approx \frac{\pi}{8}$

$\;$
$\;$
"

# ╔═╡ ac7668db-45f4-456c-a55b-b9106555fca4
md"
---

The well known but slow converging [*Leibniz* formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80) for determining $\pi$ is :

$\frac{\pi}{4} = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + ...$

$\;$
$\;$
$\;$
We partition the series into a sum of pairs:
$\;$

$\frac{\pi}{4} = \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)} = \left(1 - \frac{1}{3} \right) + \left(\frac{1}{5} - \frac{1}{7}\right) + \left(\frac{1}{9} - \frac{1}{11}\right) + ...$

$\;$
$\;$
$\;$
$\frac{\pi}{4} = \sum_{k=0,2,4,...}^\infty \left(\frac{1}{(2k+1)}-\frac{1}{2(k+1+1)} \right) = \sum_{k=0,2,4,...}^\infty \frac{2}{(2k+1)(2k+3)}$
$\;$
$\;$
$\;$
Divison by $2$ simplifies the numerator:
$\;$

$\frac{\pi}{8} = \sum_{k=0,2,4,...}^\infty \frac{1}{(2k+1)(2k+3)} = \frac{1}{1\cdot3}+\frac{1}{5\cdot7}+\frac{1}{9\cdot11}+...$

$\;$
$\;$
$\;$

"

# ╔═╡ fb0e59f2-a9c5-44c3-9d41-7cb8c0347bfb
piSum1(;a=1, b=1) =                                     # SICP, p.57, keyword parms
	>(a, b) ? 
		0 : 
		+(/(1.0, *(a, +(a, 2))), piSum1(a=+(a+4), b=b)) # keyword parameters

# ╔═╡ 2b2937e1-dba3-462c-a769-60804d676bf5
*(8, piSum1(b=1))                                       # keyword arguments

# ╔═╡ e25a03e0-886c-4862-badb-bcb5be6f477a
myError(approxPi) = abs(pi - approxPi)

# ╔═╡ 31ea7745-7d46-4ade-97d1-41ce15f970f7
myError(*(8, piSum1(b=1)))                             # error 0.4749259869231266

# ╔═╡ eb647d89-bbc8-42db-9265-5cfc9ad8a03c
*(8, piSum1(b=10))

# ╔═╡ ba2a843a-a31a-4232-8c94-3893ec0ec760
myError(*(8, piSum1(a=1, b=10)))                       # error 0.16554647754361707

# ╔═╡ b6caf296-0dea-4572-9043-d515f71d6f93
myError(*(8, piSum1(a=1, b=10^2)))                     # error 0.019998000998782572

# ╔═╡ d6144688-4c14-4db3-8c49-65c9eaf7c30b
myError(*(8, piSum1(a=1, b=10^3)))                     # error 0.0019999980000102724

# ╔═╡ 54b81d43-1219-45e9-abaa-805311268810
myError(*(8, piSum1(a=1, b=10^4)))                     # error 0.00019999999800024426    

# ╔═╡ 1745d75d-dcc8-4a32-b9a8-607f5da744d9
# stack overflow, as expected
*(8, piSum1(a=1, b=^(10, 5)))                          

# ╔═╡ ddae764c-b5c0-4eda-8c80-4f57fbeac486
# stack overflow, as expected
myError(*(8, piSum1(a=1, b=^(10, 5))))                 

# ╔═╡ 6c793213-177e-4c50-99b2-482704cbad75
md"
---
###### The *Lebesgue Measure* $\lambda(C)$ (= Length) of [*Cantor's Set (=Discontinuum) C*](https://en.wikipedia.org/wiki/Cantor_set))
(NonSICP)

We take the interval 

$C_0=[0,1]$ 

$\;$
 
$\lambda(C_0) = |C_0| = 1$

$\;$

and construct the *discontinuum* (Schilling, 2015, p.2f.) by removing ternary *open* intervals $(...,...)_3$: 

- 1st construction step generating $C_1$: Remove from $C_0$ the middle *open* interval $I_2$

$I_2=(0.1,0.2)_3=(0*3^0+1*3^{-1}, 0*3^0+2*3^{-1})=\left(\frac{1}{3},\frac{2}{3}\right)$

$\;$
$\;$

$C_1 = C_0\backslash\;I_2=\left[0,\frac{1}{3}\right]\cup\left[\frac{2}{3},1\right]$

$\;$
$\;$

$\lambda(C_1)=|C_1|=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}=\frac{\#intercals}{interval\, size}=0.\overline{6}$

$\;$
$\;$

- 2nd construction step generating $C_2$: Remove from the $2^1$ *closed* intervals of $C_1$ the middle *open* intervals $I_{t_12}$ where $t_1 \in \{0,2\}$

$\;$

$I_{02} := (0.01,0.02)_3=$
$(0*3^0+0*3^{-1}+1*3^{-2},0*3^0+0*3^{-1}+2*3^{-2})=\left(\frac{1}{9}, \frac{2}{9}\right)$

$\;$
$\;$

$I_{22} := (0.21,0.22)_3=$
$(0*3^0+2*3^{-1}+1*3^{-2},0*3^0+2*3^{-1}+2*3^{-2})=\left(\frac{7}{9}, \frac{8}{9}\right)$

$\;$
$\;$
$\;$

$C_2=C_1\;\backslash I_{02}\cup I_{22}=$

$\;$

$=\left[0,\frac{1}{9}\right]\cup\left[\frac{2}{9},\frac{1}{3}\right]\cup\left[\frac{2}{3},\frac{7}{9}\right]\cup\left[\frac{8}{9},1\right]$

$\;$
$\;$

$\lambda(C_2)=|C_2| = \frac{4}{9}=\frac{\#intercals}{interval\, size}=0.\overline{4}$

$\;$
$\;$

- 3rd construction step generating $C_3$: Remove from the $2^2$ *closed* intervals of $C_2$ the middle *open* intervals $I_{1_1t_22}$ where $t_1,t_2 \in \{0,2\}$

$\;$
$I_{002} = (0.001, 0.002)_3 =$

$\;$

$=(0*3^0 + 0*3^{-1} + 0*3^{-2} + 1*3^{-3}, 
   0*3^0 + 0*3^{-1} + 0*3^{-2} + 2*3^{-3})=$

$\;$
$\;$

$\left(0+0+0+\frac{1}{27}, 0+0+0+\frac{2}{27}\right)=\left(\frac{1}{27}, \frac{2}{27}\right) = (0.03704, 0.07407)$

$\;$
$\;$
$\;$
 
$I_{022}= (0.021, 0.022)_3=$

$\;$

$=(0*3^0 + 0*3^{-1} + 2*3^{-2} + 1*3^{-3}, 
   0*3^0 + 0*3^{-1} + 2*3^{-2} + 2*3^{-3})=$

$\;$
$\;$

$=\left(0+0+\frac{2}{9}+\frac{1}{27}, 0+0+\frac{2}{9}+\frac{2}{27}\right)=\left(\frac{7}{27}, \frac{8}{27}\right)=(0.25926,0.29630)$

$\;$
$\;$
$\;$

$I_{202}= (0.201, 0.202)_3=$

$\;$

$=(0*3^0 + 2*3^{-1} + 0*3^{-2} + 1*3^{-3}, 
   0*3^0 + 2*3^{-1} + 0*3^{-2} + 2*3^{-3})=$

$\;$
$\;$

$=\left(0+\frac{2}{3}+0+\frac{1}{27},0+\frac{2}{3}+0+\frac{2}{27}\right)=$

$\;$
$\;$

$=\left(\frac{19}{27},\frac{20}{27}\right)=(0.70370, 0.74074)$

$\;$
$\;$
$\;$

$I_{222}=(0.221, 0.222)_3=$

$\;$
$\;$

$=(0*3^0 + 2*3^{-1} + 2*3^{-2} + 1*3^{-3}, 
   0*3^0 + 2*3^{-1} + 2*3^{-2} + 2*3^{-3})=$

$\;$
$\;$

$=\left(0+\frac{2}{3}+\frac{2}{9}+\frac{1}{27},0+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}\right)=\left(\frac{25}{27},\frac{26}{27}\right)=(0.9293,0.96296)$

$\;$
$\;$
$\;$
$\;$

$C_3 = C_2 \backslash\;I_{002} \cup I_{022} \cup I_{202} \cup I_{222}=$

$\;$
$\;$

$=\left[0,\frac{1}{27}\right]\cup\left[\frac{2}{27},\frac{1}{3}\right]\cup\left[\frac{2}{9},\frac{7}{27}\right]\cup\left[\frac{8}{27},\frac{1}{3}\right]\cup\left[\frac{7}{3},\frac{19}{27}\right]\cup\left[\frac{20}{27},\frac{7}{9}\right]\cup\left[\frac{8}{9},\frac{25}{27}\right]\cup\left[\frac{26}{27},1\right]$

$\;$
$\;$

$\lambda(C_3) = \frac{8}{27}=\frac{\#intercals}{interval\, size}=0.\overline{2}\overline{9}\overline{6}.$

$\;$ 
$\;$

" 

# ╔═╡ d9eefee2-a53c-4064-8ebf-c4345a76798c
md"
###### Numeration of Intervals in the *Cantor Set*
(NonSICP)

In the $n^{th}+1$ step we remove all intervals

$I_{t_1t_2...t_{n-1}t_n2}=(0.t_1t_2...t_{n-1}t_n1, 0.t_1t_2...t_{n-1}t_n2)_3 \in \{0,2\}.$

$\;$

Endpoints are written as *triadic* numbers with numbers $t_1t_2...t_{n-1}t_n \in \{0,2\}.$ The sequence of triadic numbers $t_1...t_n$ encodes the position of the interval $I_{t_1t_2...t_{n-1}t_n2}$ in relation to the interval $I_{t_1t_2...t_{n-1}2}.$ $t_n=0$ means that the interval is *left* of $I_{t_1t_2...t_{n-1}2}$ and $t_n=2$ means that it is *right* of. This results in a binary tree (plot below).

The *Lebesgue* measure (= length) of the interval $I_{t_1t_2...t_{n-1}t_n2}$ is:

$\;$

$\lambda(I_{t_1t_2...t_{n-1}t_n2})=0.t_1t_2...t_{n-1}t_n2-
0.t_1t_2...t_{n-1}t_n1=0.00...001_3=3^{-(n+1)}.$

$\;$
$\;$

$\text{e.g. }3^{-(0+1)}=\frac{1}{3},\; 3^{-(1+1)}=\frac{1}{9},\; 3^{-(2+1)}=\frac{1}{27},\;...$  

$\;$
$\;$

So interval lengths can be derived from this ternary encoding:

- 1st step: 2^0 intervals $\lambda(I_2) = 0.2_3-0.1_3=0.1_3=3^{-(0+1)}=\frac{1}{3}$
- 2nd step: 2^1 intervals $\lambda(I_{02}), \lambda(I_{22}) =3^{-(1+1)}=\frac{1}{9}$
- 3rd step: 2^2 intervals $\lambda(I_{002}), \lambda(I_{022}), \lambda(I_{022}) , \lambda(I_{222})=3^{-(2+1)}=\frac{1}{27}$
- (n+1)th step: 2^n intervals with length $3^{-(n+1)}.$

$\;$

" 

# ╔═╡ b61f0964-d023-4bba-9822-812e3218f716
begin
plot(x->1.0, 0.0, 1.0, size=(700, 400), xlims=(-0.02, 1.24), ylims=(0, 1.2), line=:blue, lw=3, label="", title="Construction of Cantor's Set (=Discontinuum)", framestyle=:semi)
plot!([(0.0,1.02),(0.0,0.98)],line=:blue,lw=1,label="") # left vertical start bar
plot!([(1.0,1.02),(1.0,0.98)],line=:blue,lw=1,label="") # right vertical end bar
annotate!(1.1, 1.0, L"C_0")
#----------------------------------------------------------------------------------
annotate!(1/3, 0.83, L"\frac{1}{3}", 10); annotate!(2/3, 0.83, L"\frac{2}{3}", 10)
plot!(x->0.75, [[0.0, 1/3], [2/3, 1.0]], line=:blue, lw=3, label="")
plot!([(0.0,0.77),(0.0,0.73)],line=:blue,lw=1,label="") # left vertical start bar
plot!([(1/3,0.77),(1/3,0.73)],line=:blue,lw=1,label="") # left vertical bar
plot!([(2/3,0.77),(2/3,0.73)],line=:blue,lw=1,label="") # left vertical bar
plot!([(1.0,0.77),(1.0,0.73)],line=:blue,lw=1,label="") # right vertical end bar
annotate!(1.1, 0.75, L"C_1")
annotate!(1/2, 3/4, L"I_2")
annotate!(1/9, 0.58, L"\frac{1}{9}", 10); annotate!(2/9, 0.58, L"\frac{2}{9}", 10)
annotate!(7/9, 0.58, L"\frac{7}{9}", 10); annotate!(8/9, 0.58, L"\frac{8}{9}", 10)
#----------------------------------------------------------------------------------
plot!(x->0.50, [[0.0, 1/9], [2/9, 3/9], [6/9, 7/9], [8/9, 9/9]], line=:blue, lw=3, label="")
plot!([(0.0,0.52),(0.0,0.48)],line=:blue,lw=1,label="") # left vertical start bar
plot!([(1/9,0.52),(1/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(2/9,0.52),(2/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(3/9,0.52),(3/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(6/9,0.52),(6/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(7/9,0.52),(7/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(8/9,0.52),(8/9,0.48)],line=:blue,lw=1,label="") # left vertical bar
plot!([(1.0,0.52),(1.0,0.48)],line=:blue,lw=1,label="") # left vertical bar
annotate!(1.1, 0.5, L"C_2")
annotate!(0.17, 0.5, L"I_{02}")
annotate!(0.835, 0.5, L"I_{22}")
#----------------------------------------------------------------------------------
plot!(x->0.25, [[0.0, 1/27], [2/27, 3/27], [6/27, 7/27], [8/27, 9/27], [18/27, 19/27], [20/27, 21/27], [24/27, 25/27], [26/27, 27/27]], line=:blue, lw=3, label="")
plot!([( 0/27,0.27),( 0/27,0.23)],line=:blue,lw=1,label="") # left vertical start bar
plot!([( 1/27,0.27),( 1/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 2/27,0.27),( 2/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 3/27,0.27),( 3/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 6/27,0.27),( 6/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 7/27,0.27),( 7/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 8/27,0.27),( 8/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([( 9/27,0.27),( 9/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(18/27,0.27),(18/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(19/27,0.27),(19/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(20/27,0.27),(20/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(21/27,0.27),(21/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(24/27,0.27),(24/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(25/27,0.27),(25/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(26/27,0.27),(26/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
plot!([(27/27,0.27),(27/27,0.23)],line=:blue,lw=1,label="") # left vertical bar
annotate!(1.1, 0.25, L"C_3")
annotate!(0.075, 0.20, L"I_{002}")
annotate!(1/27-0.005, 0.32, L"\frac{1}{27}",8), annotate!(2/27+0.007, 0.32, L"\frac{2}{27}",8)
annotate!(7/27-0.005, 0.32, L"\frac{7}{27}",8), annotate!(8/27+0.007, 0.32, L"\frac{8}{27}",8)
annotate!(19/27-0.005, 0.32, L"\frac{19}{27}",8), annotate!(20/27+0.007, 0.32, L"\frac{20}{27}",8)
annotate!(25/27-0.005, 0.32, L"\frac{25}{27}",8), annotate!(26/27+0.007, 0.32, L"\frac{26}{27}",8)
annotate!(0.3, 0.20, L"I_{022}")
annotate!(0.745, 0.20, L"I_{202}")
annotate!(0.97, 0.20, L"I_{222}")
#----------------------------------------------------------------------
# binary tree
plot!([(1/2, 3/4), (0.17, 0.5)], color=:pink,lw=2,ls=:dash,label="")
plot!([(1/2, 3/4),(0.835, 0.5)], color=:pink,lw=2,ls=:dash,label="")
plot!([(0.17, 0.5),(0.055, 0.25)], color=:pink,lw=2,ls=:dash,label="")
plot!([(0.17, 0.5),(0.28, 0.25)], color=:pink,lw=2,ls=:dash,label="")
plot!([(0.835, 0.5),(0.722, 0.25)], color=:pink,lw=2,ls=:dash,label="")
plot!([(0.835, 0.5),(0.945, 0.25)], color=:pink,lw=2,ls=:dash,label="")
#----------------------------------------------------------------------
end # begin

# ╔═╡ d2f67bf1-a301-4c93-aaac-4c66a7c9b944
md"
---
###### Measuring $\lambda(C_n)$ 
(NonSICP)

The *Lebesgue Measure* $\lambda(C)$ of *Cantor's Discontinuum* $C$ is (c.f. Schilling, 2015, p.2f.):

$\lambda(C_0)=\lambda([0,1])=1$

$\;$

$\lambda(C_1)=\lambda(C_0)-\frac{1}{3}=\lambda([0,1])-\frac{1}{3}=1-\frac{1}{3}= \frac{2}{3}=0.\overline{6}$

$\;$
$\;$

$\lambda(C_2)=\lambda(C_1)-2\cdot\frac{1}{9} = \frac{2}{3}-\frac{2}{9}=\frac{6}{9}-\frac{2}{9}=\frac{4}{9}=0.\overline{4}$

$\;$
$\;$

$\lambda(C_3)=1-\frac{1}{3}\left[\frac{2^0}{3^0}+\frac{2^1}{3^1}+\frac{2^2}{3^2}\right]=1-\frac{1}{3}\left[1+\frac{2}{3}+\frac{4}{9}\right]=$

$\;$
$\;$
$\;$

$=1-\frac{1}{3}\cdot\frac{19}{9}=1-\frac{19}{27}=\frac{8}{27}=0.\overline{2}\overline{9}\overline{6}$

$\;$
$\;$
$......$
$\;$

$\lambda(C_{n+1})=1-2^0\times\frac{1}{3^1}-2^1\times\frac{1}{3^2}-...-2^n\times\frac{1}{3^{n+1}}=1-\frac{1}{3}\sum_{j=0}^n\frac{2^j}{3^j}=1-\frac{1}{3}\sum_{j=0}^n\left(\frac{2}{3}\right)^j$

$\;$
$\;$
$......$
$\;$

$\lambda(C)=1-\sum_{n=0}^\infty\frac{2^n}{3^{n+1}}=1-\frac{1}{3}\sum_{n=0}^\infty\frac{2^n}{3^n}=1-\frac{1}{3}\sum_{n=0}^\infty\left(\frac{2}{3}\right)^n=0.$

$\;$
$\;$
$\;$

This means that the *Cantor set* is a [*Null Set*](https://en.wikipedia.org/wiki/Null_set). It has length *zero* though it is not empty. It contains at least the endpoints of the intervals.

$\;$

"

# ╔═╡ 4dae9415-e8e2-4133-9cef-a3c4618bdbda
md"
Because $\sum_{n=0}^\infty\left(\frac{2}{3}\right)^n$ is a convergent geometric series with $r=\frac{2}{3}$ the sum is 

$\;$
$\;$

$\sum_{n=0}^\infty\left(r\right)^n=\frac{1}{1-r}$

$\;$
$\;$

and

$\;$

$\lambda(C)=1-\frac{1}{3}\sum_{n=0}^\infty\left(\frac{2}{3}\right)^n=1-\frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right)=1-\frac{1}{3}\left(\frac{1}{\frac{1}{3}}\right)=1-\frac{1}{3}\left(\frac{3}{1}\right)=0$

$\;$
$\;$
$\;$
$\;$

*C* is not empty (e.g. Schilling, 2015, p.3). At least the end points of intervals are left over in $C$ after removings.
$\;$

"

# ╔═╡ ed9af448-5867-4cfd-815a-c295eaffac8b
function λC(n; j=0, accu=0.0)
	r = 2/3
	if j == n
		1 - 1/3 * accu
	else
		reduction = r^j
		accuOld = accu
		accuNew = accu + reduction
		if accuOld ≈ accuNew
			error("no substantial reduction with j = $j")
		end # if
		λC(n; j=j+1, accu=accu+reduction)
	end # if 
end # function λC 

# ╔═╡ ed8e7ccf-5c04-4304-baf9-60cc07773d28
λC(0)                      # ==> 1.0  --> :)

# ╔═╡ 83b7e51e-2e73-4448-8cb1-ed98787c3994
λC(1)                      # ==> 2/3  --> 0.666... --> :)

# ╔═╡ 14b8a22d-5443-4fae-abd9-b54a283eaa92
λC(2)                      # ==> 4/9  --> 0.444... --> :)

# ╔═╡ c9905375-c8ea-4403-8ce7-2af82324027a
λC(3)                      # ==> 8/27 --> 0.296... --> :)

# ╔═╡ 6052b447-cd20-4f1e-87e1-bda2ea6461a3
λC(10)

# ╔═╡ 650c0387-e516-4b08-8c9a-aa49682183a4
λC(50)                    # ==> no substantial reduction with j = 42

# ╔═╡ 299fb3ee-a3a6-44d1-9b0c-094283e6dfa4
λC(42)

# ╔═╡ f89d9a5d-0fd8-4684-82bd-797de43582a2
λC(43)                     # ==> no substantial reduction with j = 42

# ╔═╡ c193805e-df87-4842-bdb5-d1991db9ccf0
function λC2(n) #; j=0, accu=0.0)
    j = 0
	if n == 0
		1
	else
		r = 2/3 
		1 - 1/3 * sum(
			j -> r^(j-1), [j for j in 1:n], init=0.0)
	end # if
end # function λC 

# ╔═╡ 70ec1fa0-2e9d-4aab-923a-10fa0fb249a5
λC2(0)

# ╔═╡ 2eba09ba-f280-47c6-8753-18b9a77c4a5b
λC2(1)

# ╔═╡ 5d34717f-f9fe-4796-9fc5-51462bdabe85
λC2(2)

# ╔═╡ 6f6e5499-7dcb-4a3a-b0b9-a8ca9898b689
λC2(3)

# ╔═╡ b37131ce-d9c0-4b9e-9392-f752dba44c13
λC2(42)                       # same as λC(42)

# ╔═╡ 2874445b-ced9-45bd-9094-5a1e48651831
λC2(43)                       # less than λC(43)

# ╔═╡ f9034ab8-2af6-4d79-be1d-1da8c6cccd81
λC2(50)                       # less than λC(50)

# ╔═╡ 477debe7-5232-4bcf-8109-3ee8352f0047
λC2(60)                       # less than λC(60)

# ╔═╡ 2c0a3d9d-8656-46b9-83ce-080854f59923
λC2(70)                       # less than λC(70)

# ╔═╡ 281ea205-6462-45bb-a63a-24de59f396e2
λC2(70) 

# ╔═╡ cb608af4-73cf-4c5b-824d-073b36414664
λC2(80)

# ╔═╡ e566a3ee-1f78-4caa-b56e-0c6354074fec
λC2(85)

# ╔═╡ bf31819b-32ef-4099-bd68-982f085be0c4
λC2(87)

# ╔═╡ 0d76bef3-53bf-4947-862b-8f46a0438481
λC2(88)

# ╔═╡ 07577047-ac4f-4f8f-950d-2db31ee94539
λC2(89)

# ╔═╡ e846e3a1-0b2f-43e0-adfb-866d9511c4f2
λC2(90)                         # no further improvement observable

# ╔═╡ fdcd0ad9-a81c-48c8-ab7d-8f8b70e38c4d
md"
---
(SICP, p.58)

$sum : (\mathbb N \rightarrow \mathbb N) \times (\mathbb N \cup \mathbb R) \times (\mathbb N \rightarrow \mathbb N) \times (\mathbb N \cup \mathbb R) \rightarrow (\mathbb N \cup \mathbb R)$

$\;$
$\;$

$(f, a, succ, b) \mapsto sum(f, a, succ, b)$

$\;$

$sum(f, a, succ, b) := \sum_{i=a}^b f(i) = f(a) + f(succ(a)) + ... + f(b)$

$\;$
$\;$
$\;$
"

# ╔═╡ da24559b-1937-4258-847d-e0f7b7a7c293
function sum1(f, a, succ, b)                                # SICP, p.58
	if >(a, b)
		0
	else
		+(f(a), sum1(f, succ(a), succ, b))
	end # if
end # function sum1

# ╔═╡ b9c17500-0301-4a9f-8707-c18fda55261c
inc(n) = +(n, 1)                                            # SICP, p.58

# ╔═╡ 76174121-4f07-4e04-9bde-f98344fc1a1e
sumCubes2(a, b) = sum1(cube, a, inc, b)                     # SICP, p.59

# ╔═╡ 792c2312-2469-4844-b7b8-4acd42333a82
sumCubes2(0, 1) 

# ╔═╡ 39a8a3f7-391d-4adf-a217-9ce1db164099
# 0 + 1 + 2*2*2 + 3*3*3 = 1 + 8 + 27 = 36
sumCubes2(0, 3)                        

# ╔═╡ eb294e7b-3eb4-4038-bf10-3870819dd2ae
sumCubes1(1, 10)                                            # SICP, p.59

# ╔═╡ 0779063c-bb2c-4060-84d7-e6268f9255a2
identity(x) = x                                             # SICP, p.59

# ╔═╡ d14308c5-374a-495d-8865-1f52e198fb5e
sum_integers2(a, b) = sum1(identity, a, inc, b)             # SICP, p.59

# ╔═╡ d195654c-302f-4c80-9d4b-0d95978b6a7c
sum_integers2(0, 6)

# ╔═╡ 33ea2359-475f-42ca-8de4-5f4e91148529
sum_integers2(1, 10)                                        # SICP, p.59

# ╔═╡ 99baa676-35fc-4bcb-bd81-9f86b154c307
md"
---
###### Alternative $\pi_{sum_2}$ with functional arguments $term$ and $next$ 
(SICP, p.59)
"

# ╔═╡ 2a14bebd-8b53-442a-965f-983f040eeea3
md"
---
###### Riemann Integration by the [Trapezoidal Method](https://en.wikipedia.org/wiki/Trapezoidal_rule)
(SICP, p.59)
$\;$

$\int_a^b f(x)\; dx \approx \left[f\left(a+\frac{Δx}{2}\right)+f\left(a+Δx+\frac{Δx}{2}\right)+f\left(a+2Δx+\frac{Δx}{2}\right)...\right]\;Δx$
$\;$
$\;$
$\;$
$=\left[\sum_{i=0}^{b/Δx}f\left(a + i \cdot Δx + \frac{dx}{2}\right)\right]\;Δx$

$\;$
$\;$
$\;$
;
"

# ╔═╡ d0c513b0-e921-45b2-a12b-1ad41a1e902e
function integral1(f, a, b; Δx=0.01)                               # SICP, p.60
	add_Δx(x) = x + Δx
	sum1(f, a + Δx/2.0, add_Δx, b) * Δx
end # function integral1

# ╔═╡ cd4ad65b-6fa0-4eda-8dd5-0bb3d04eeeb2
md"
---
###### Cubic function 
(SICP, p.60)

$$\int_0^1 x^3 dx= \left.\frac{x^4}{4}\right|_0^1=\frac{1^4}{4}-\frac{0^4}{4}=\frac{1}{4}$$
$\;$
$\;$
$\;$
$\;$
"

# ╔═╡ 02bf20ff-0e44-40bc-bd6a-1e55e2086571
plot(cube, 0.0, 1.0, size=(525, 400), xlims=(0.0, 1.1), ylims=(0, 1.2), line=:darkblue, fill=(0, :lightblue), title=L"$x \mapsto x^3$", framestyle=:semi, label="")

# ╔═╡ bf281edd-d933-446b-8b54-903e52c4505b
integral1(cube, 0, 1)                                  # SICP, p.60

# ╔═╡ dc5ec344-9513-4cda-b97f-511a7449cb6d
integral1(cube, 0, 1, Δx=0.001)                        # SICP, p.60

# ╔═╡ 71319af0-3051-41c8-8c12-20c142901c11
integral1(cube, 0, 1, Δx=1.0E-4)

# ╔═╡ 7dffb4c0-1820-46c4-8676-f327de55048e
md"""
---
###### Halfparabola
(NonSICP), Stark, P.A., *Introduction to Numerical Methods*, 1970, ch. 6.2-6.3, p. 196, Example 6-1
$\;$

$\int_0^1\left(6-6x^5\right)dx = \left.\left(6x-6\frac{x^6}{6}\right)\right|_0^1=(6-1)=5$
$\;$
$\;$
$\;$
$\;$

"""

# ╔═╡ 76170408-797d-451b-bcf6-b69fec927085
halfParabola(x) =  -6x^5 + 6 # Stark, P.A., Intro to Num. Methods, 1970, p.196f.

# ╔═╡ b2db2a4a-8a5e-43b1-bb99-d37828dbec0c
plot(halfParabola, 0.0, 1.0, size=(200, 400), xlim=(0.0, 1.1), ylim=(0, 6.5), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title=L"$x\mapsto -6x^5 + 6$", label="")

# ╔═╡ 73a0deb5-6dc3-41d8-b810-5553c74455ed
integral1(halfParabola, 0.0, 1.0, Δx=0.1)          # should be 5.0

# ╔═╡ 08f1b671-dc0a-4b7a-92ce-4568c24589f1
integral1(halfParabola, 0.0, 1.0, Δx=0.01)         # should be 5.0

# ╔═╡ 2cf94a78-f3ea-4e02-b417-aaa6510ff328
md"
---
###### [Standard Normal (= Gaussian) Distribution](https://en.wikipedia.org/wiki/Normal_distribution)
(NonSICP)
"

# ╔═╡ b2e9582d-bf46-486b-b7fb-1e3a68e77efe
gaussianDensity(x; μ=0.0, σ=1.0) = 1/(σ*sqrt(2π))*exp(-(1/2)*((x-μ)/σ)^2)

# ╔═╡ bd63330c-e1f8-4f4e-836f-b1285539a0c1
let x1 = +1.0
	x2 = -1.0
	y1 = gaussianDensity(+1.0)
	y2 = gaussianDensity(-1.0)
	plot(gaussianDensity, -1.0, 1.0, size=(525, 400), xlim=(-3.0, 3.0), ylim=(0, 0.45), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title=L"$p(-1.0 < X < +1.0)$ for $X \sim N(X\;\;| μ=0.0, σ=1.0)$", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"f(X)")
	plot!(gaussianDensity, -3.0, -1.0,  line=:darkblue, label=L"$f(X)$")
	plot!(gaussianDensity, 1.0, 3.0,  line=:darkblue, label=L"$f(X)$")
	plot!([x1, x1], [0, y1], color=:red, label=L"$x=+1.0=\sigma$")
	plot!([x2, x2], [0, y2], color=:red, label=L"$x=-1.0=-\sigma$")
end # let

# ╔═╡ 12e0fa05-135d-48b6-b6c2-c28ce006b2d9
integral1(gaussianDensity, -1.0, 1.0, Δx=0.1)         # ==> p = 0.682

# ╔═╡ 23824efd-d3d8-4e02-8ddd-5e20da7e99a4
integral1(gaussianDensity, -1.0, 1.0)                 # ==> p = 0.682

# ╔═╡ ae961cd1-bab0-41e2-9bc5-918d7346f82d
md"
---
##### 1.3.1.2 idiomatic *imperative* Julia ...
###### ... with *abstract* types 'Real', 'Integer', *self-defined*, 'while', '+='
" 

# ╔═╡ cb1cbf7f-98e0-49d4-b12e-ee8b8808d316
pi                                                           # Julia constant π

# ╔═╡ d0ea0cc2-fdc3-4055-a651-946a66cb7db0
typeof(pi)                                                   # Irrational !

# ╔═╡ 9fde9515-3848-44c1-b33b-de9bae9c77c6
subtypes(Real)

# ╔═╡ 890ab8e8-12df-4975-90a4-fda312d60ba2
# idiomatic Julia-code with 'Real', 'while', '+='
#-----------------------------------------------------
function sumIntegers2(a::Real, b::Real)::Real
	sum = 0
	while !(a > b)
		sum += a
		a += 1
	end # while
	sum
end # function sumIntegers2

# ╔═╡ 20c6c63f-6f04-4aa2-987a-e6789a916383
sumIntegers2(1, 10)

# ╔═╡ 38295b22-5dfb-4e3a-bdd7-e7b91c65d342
sumIntegers2(2, 10)                      # ==> Integer

# ╔═╡ 46466ddf-7b65-49f0-b8d0-d0039f1165e0
sumIntegers2(2., 10)                     # ==> Float

# ╔═╡ 76cbceaa-ff7b-4e49-8935-6eda98b68729
sumIntegers2(2, 10.)                     # ==> Integer

# ╔═╡ 8a4adbdc-2553-4f6a-8cb2-e0c3d35cca08
sumIntegers2(2., 10.) 

# ╔═╡ 81c0cc23-68f9-4d91-af01-38898e6d50af
# idiomatic Julia-code with 'Real, 'while', '+='
#---------------------------------------------------
function sumCubes3(a::Real, b::Real)::Real
	#-----------------------------------------------
	cube(x) = x^3
	#-----------------------------------------------
	sum = 0
	while !(a > b)
		sum += cube(a)
		a += 1
	end # while
	sum
end # function sumCubes3

# ╔═╡ 732458d7-1938-4be1-9de2-8071d1ad644d
sumCubes3(0, 1)

# ╔═╡ 39c2206d-8a4d-49d0-a8d5-c7a744443000
sumCubes3(0, 3)                            # 0 + 1 + 2*2*2 + 3*3*3 = 1 + 8 + 27 = 36

# ╔═╡ 4d0c6403-7588-4971-8046-1f99e3a421a6
sumCubes3(0, 4)                            # sumCubes3(0, 3) + 4^3 = 36 + 64 = 100

# ╔═╡ 1e034229-e2a0-420e-97e7-8a02a577fe5a
sumCubes1(1, 4)                            # 36 + 4^3 = 36 + 64 = 100

# ╔═╡ 69618ff5-44f5-4d60-8297-cc15457a2ed8
# idiomatic Julia-code with 'Real','while', '+='
#-------------------------------------------------------
function piSum3(a::Real, b::Real)::Real
	sum = 0
	while !(a > b)
		sum += 1.0 / (a * (a + 2))  
		a += 4
	end # while
	sum
end # function piSum3

# ╔═╡ 3127686e-0c30-48e2-8c73-e7563ee60124
8 * piSum3(1, 10^5)

# ╔═╡ 5a847c91-d008-434c-98de-90a74335012c
myError(8 * piSum3(1, 10^5))

# ╔═╡ 9fbc3781-1cf3-4b94-bc93-b5d8c7e0ad24
myError(8 * piSum3(1, 10^8))

# ╔═╡ 23cf565a-21a1-407d-981d-1ddc433327a8
8 * piSum3(1, 10^10)

# ╔═╡ 5b4e0581-36b1-47d3-be06-a84cfee8788f
myError(8 * piSum3(1, 10^10))

# ╔═╡ 3a72d464-a12c-4de5-8c8c-7768c60bec3c
typeof(Function)

# ╔═╡ 28aab614-3469-43d8-ab48-6feb0d30be92
typeof(DataType)

# ╔═╡ 82a083c1-2637-49cc-ada1-2c006c26b39f
supertype(Function)

# ╔═╡ a9f09569-210d-4bbd-8b81-556f628d9220
subtypes(Function)

# ╔═╡ d4290924-cbe3-4f51-88f7-3aaacdfd6b4f
# idiomatic Julia-code with 'Function', 'Real', 'while', '+='
#------------------------------------------------------------------
function sum2(f::Function, a::Real, succ::Function, b::Real)::Real
	sum = 0
	while !(a > b)
		sum += f(a)
		a = succ(a)
	end # while
	sum
end # function sum2

# ╔═╡ 7dc42461-ba3a-4837-ba37-e91920c8dc66
function piSum2(a::Real, b::Real)::Real                     # SICP, p.59
	#---------------------------------------------------------------------
	piTerm(x) = 1.0 / (x * (x + 2))
	piNext(x) = x + 4
	#---------------------------------------------------------------------
	sum2(piTerm::Function, a::Real, piNext::Function, b::Real)
end # unction piSum2

# ╔═╡ 7fcba949-68ad-4622-8492-a21c21c81d56
8 * piSum2(1, 10^3)                                         # SICP, p.59

# ╔═╡ 254b9112-0f77-4919-9fdb-1e8d4785187b
8 * piSum2(1, 10^6)

# ╔═╡ 5967b1d1-0990-4ebc-ae8b-6e6ec00b4f5d
8 * piSum2(1, 10^6)

# ╔═╡ ba73b5bb-e803-4e5a-91d9-9c93da299930
8 * piSum2(1.0, 10.0^6)

# ╔═╡ 0ad7a6d9-1d24-41dc-a573-a8856dfef417
sum_cubes4(a, b) = sum2(cube, a, inc, b)

# ╔═╡ f2d4ee8f-a71e-4370-845f-ac3d37cb4a51
sum_cubes4(1, 1)

# ╔═╡ fd9dadfb-e205-4016-9bad-ceb7fcc1f417
sum_cubes4(1, 2)

# ╔═╡ 9636b204-dc6b-4e9c-88b4-7f4f3a0ff353
sum_cubes4(1, 10)

# ╔═╡ 13f1da45-78ce-4ee9-86ed-0bdb4e592f1c
function integral2(f, a, b; Δx=0.01) 
	add_Δx(x) = x + Δx
	sum2(f::Function, (a + Δx/2.0), add_Δx, b) * Δx
end

# ╔═╡ 7174bf72-b914-4944-95ac-a47cc7ddbfef
integral2(halfParabola, 0, 1, Δx=0.01)             # should be 5.0

# ╔═╡ 90d3e292-ecdf-459f-b451-a5fabc6fe755
integral2(halfParabola, 0, 1, Δx=1.0E-3)           # should be 5.0

# ╔═╡ b94bf716-f77f-4127-aaa8-2b2f52fab923
integral2(halfParabola, 0, 1, Δx=1.0E-4)           # should be 5.0

# ╔═╡ df78019b-24be-48d2-b675-d656447a1c0d
5.0 - integral2(halfParabola, 0.0, 1.0, Δx=1.0E-5) # should be 5.0

# ╔═╡ 9cf19259-52ce-46a7-8a27-89e26c7d5ca4
5.0 -integral2(halfParabola, 0.0, 1.0, Δx=1.0E-6)  # should be 5.0

# ╔═╡ bc194820-97b7-4836-bc48-340e637e7f0d
5.0 -integral2(halfParabola, 0.0, 1.0, Δx=1.0E-7)  # should be 5.0

# ╔═╡ 40e24f4d-9966-4984-b255-34c6cd3e0d7e
integral2(gaussianDensity, -1.0, 1.0, Δx=1.0E-3)      # ==> p = 0.682

# ╔═╡ 63991eb0-ef07-471e-89df-97b9e5b95b19
integral2(gaussianDensity, -1.0, 1.0, Δx=1.0E-5)      # ==> p = 0.682

# ╔═╡ 43b2d09b-8822-46bc-9372-6ee4b9f1141a
integral2(cube, 0, 1)

# ╔═╡ fba935d4-ee66-44b8-8e5b-bf14a8c08810
integral2(cube, 0.0, 1.0, Δx=1.0E-4)

# ╔═╡ f7839cb5-a075-4e70-9cca-96dd760d356f
md"
---
###### Lebesgue Integration
(nonSICP)

This section contains a Julia-scripts not found elsewhere. The reason for this may be the fact that for most practical purposes numerical integration a la *Riemann* would do. The superiority of Lebesgue integration shines in more theoretical contexts when Riemann integration fails (e.g. integral of the [Dirichlet function](https://en.wikipedia.org/wiki/Dirichlet_function)).

The difference between both methods is:
*'... the fact that the Riemann sums partition the domain of the function without taking into account the shape of the function, thus slicing up the area under the function $vertically$. Lebesgue's approach is exactly the opposite: the domain is partitioned according to the values of the function at hand, leading to a $horizontal$ decomposition of the aera'* (Schilling, 2005, p.94).

Our Julia scripts $lebesgueIntegral1, lebesgueIntegral2$ sum *horizontal sclices* according to the formula:

$\int_a^b f(x)\;dx = \lim_{\Delta y \to 0} Δy\; \left[\sum_{y=0}^{max(f(x))/Δy} \mu(\{x|f(x) > (y + \Delta y)\})\right]$

$\;$
$\;$
$\;$

Compared to the Riemannian approach this script lacks efficiency. This is due to the high number of calls of $f(x)$ in each summand. We can improve efficiency slightly by caching function calls.
"

# ╔═╡ a9ee6052-2021-4543-8680-b66f76b659af
#------------------------------------------------------------------------------------
# lebesgueIntegral1 only integrates *positive* functions
# - integration method is summation of horizontal slabs
# - recursive function earchForMax
#------------------------------------------------------------------------------------
function lebesgueIntegral1(f::Function; a=0.0, b=1.0, y=0.0, Δx=0.01) 
	#--------------------------------------------------------------------------------
	Δy = Δx
	add_Δx(x) = x + Δx
	add_Δy(y) = y + Δy	#--------------------------------------------------------------------------------
	function μ(y)
		let μΔx(x) = (f(x) > add_Δy(y)) ? Δx : 0
			sum(μΔx(x) for x in a:Δx:b)
		end # let
	end # function μ
	#--------------------------------------------------------------------------------
	function searchForMax(f, fmax, x, b) 
		if add_Δx(x) > b
			fmax
		else
			let x = add_Δx(x)
				fnew = f(x)                                  # new tentative maximum
				if fmax < fnew 
					begin
						fmax = fnew
						searchForMax(f, fmax, x, b)
					end
				else
					searchForMax(f, fmax, x, b)
				end # if
			end # let
		end # if
	end # function searchForMax
	#--------------------------------------------------------------------------------
	fmax = searchForMax(f, f(a), a, b)   # the search for maximum of function f
	# vertical sum of measures μ of horizontal slabs
	sum2(μ::Function, 0::Real, add_Δy::Function, (fmax-Δy)::Real) * Δy 
	#--------------------------------------------------------------------------------
end # function lebesgueIntegral1

# ╔═╡ a6c249cf-dd5f-4a1e-a4b7-beb30ea16fa5
unitStep(x) = 0 < x ≤ 1 ? 1 : 0

# ╔═╡ f6634d4a-06ac-4b83-a94a-c798473cc067
plot(unitStep, -0.5, 1.5, size=(525, 400), xlim=(-0.5, 1.5), ylim=(0, 1.5), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title=L"unitStep(x) = 0 < x\; ≤ \;1 \;? \; 1 : 0", label="")

# ╔═╡ 010ca46b-f127-4b26-99e4-e96f5f4d7c6e
lebesgueIntegral1(unitStep, Δx=0.1E-3)

# ╔═╡ b8ed62d0-1e0b-478e-8034-7d06e3d41db1
 # stack overflow, as expected
lebesgueIntegral1(unitStep, Δx=0.1E-4)           

# ╔═╡ d6cc5ab7-376a-4e3a-a0c1-edb0b76b4e45
unitPyramid(x) = x ≤ 1/2 ? x : 1 - x

# ╔═╡ 8f84149d-ea67-421c-ae56-82748dbe5421
plot(unitPyramid, -0.1, 1.1, size=(525, 400), xlim=(-0.1, 1.1), ylim=(0, 0.6), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title=L"unitPyramid(x) = x\; ≤\; 1/2 \;?\; x : 1 - x", label="")

# ╔═╡ 000ec5a2-6963-4ac3-870a-a7fd5bad4f6c
md"
###### Area of Pyramid: $=\frac{length*height}{2}=(1 * 1/2)/2 = 1/4$
"

# ╔═╡ 24fe641f-05e5-4bd2-b109-6105d6df90ea
lebesgueIntegral1(unitPyramid, Δx=0.1E-3)     # (1 * 1/2)/2 = 1/4

# ╔═╡ a161ea4f-74b4-4a0b-bce6-fb435685ba91
errorPyramid1 = abs(1/4-lebesgueIntegral1(unitPyramid, Δx=0.1E-3))

# ╔═╡ f2b81dfd-caa8-4d22-ac4a-662694b093a8
# stack overflow, as expected
lebesgueIntegral1(unitPyramid, Δx=0.1E-4)       

# ╔═╡ 2897cda1-e00b-4bbc-bed3-fcc3e01e5203
lebesgueIntegral1(halfParabola, Δx=0.1E-3)  

# ╔═╡ 9c86d18f-5a34-4c2c-8290-cd9b33cb887d
lebesgueIntegral1(gaussianDensity, a=-1.0, Δx=0.1E-2)

# ╔═╡ 96e1b398-2233-4494-a9cf-aac19a418014
# stack overflow, as expected
lebesgueIntegral1(gaussianDensity, a=-1.0, Δx=0.1E-3)

# ╔═╡ 5efe1b56-4f21-4c3c-8b27-fdaaa90a558a
md"
---
###### Lebesgue Integration with $while$-Loops
"

# ╔═╡ 234a5247-a16b-4d4c-a47b-06c832391e60
#------------------------------------------------------------------------------------
# lebesgueIntegral2 only integrates *positive* functions
# - integration method is summation of horizontal slabs
# - *non*recursive function earchForMax using *while*
#------------------------------------------------------------------------------------
function lebesgueIntegral2(f::Function; a=0.0, b=1.0, y=0.0, Δx=0.01) 
	#--------------------------------------------------------------------------------
	Δy = Δx
	add_Δx(x) = x + Δx
	add_Δy(y) = y + Δy	#--------------------------------------------------------------------------------
	function μ(y)
		let μΔx(x) = (f(x) > add_Δy(y)) ? Δx : 0
			sum(μΔx(x) for x in a:Δx:b)
		end # let
	end # function μ
	#--------------------------------------------------------------------------------
	function searchForMax(f, fmax, x, b) 
		xmax = x
		while !(add_Δx(x) > b)
			x = add_Δx(x)
			fnew = f(x)                           # new tentative maximum
			if 	fmax < fnew 
				begin
					fmax = fnew
					xmax = x
				end # begin
			end # if
			x = add_Δx(x)
		end # while
		fmax
	end # function searchForMax
	#--------------------------------------------------------------------------------
	fmax = searchForMax(f, f(a), a, b)   # the search for maximum of function f
	# vertical sum of measures μ of horizontal slabs
	sum2(μ::Function, 0::Real, add_Δy::Function, (fmax-Δy)::Real) * Δy 
	#--------------------------------------------------------------------------------
end # function lebesgueIntegral2

# ╔═╡ 7276b63c-f0aa-4cfc-9014-ba5cee36e9fa
lebesgueIntegral2(unitPyramid, Δx=0.1E-3)

# ╔═╡ a31a361f-fcc7-4824-bcda-627760d5474a
abs(1/4 - lebesgueIntegral2(unitPyramid, Δx=0.1E-3))         # ==> 1/4 --> :)

# ╔═╡ 3725189a-4eb5-4fc1-bb73-8d0976ef0e88
# *no* stack overflow as above with version 1
lebesgueIntegral2(unitPyramid, Δx=0.1E-4)   

# ╔═╡ ebfe529d-19ec-415a-a5d4-9183652c26cd
abs(1/4 - lebesgueIntegral2(unitPyramid, Δx=0.1E-4))

# ╔═╡ 926772b6-fc86-4728-9d9d-f4bcc41ef9b8
# integral2Gaussian = 0.6826819276010768
lebesgueIntegral2(gaussianDensity, a=-1.0, Δx=0.1E-4)   # ==> 0.6826819276010768

# ╔═╡ 93b4f27b-7c83-4d61-bb82-b0651f9385cf
md"
---
Our Julia script $lebesgueIntegral3$ sum *steps* according to the formula:

$\int_a^b f(x)\;dx = \int_a^b f(x)\;\mu(dx) = \lim_{\Delta y \to 0} \left[\sum_{y=0}^{max(f(x))/Δy} y\cdot \lambda(\{x|(y + \Delta y) > f(x) \geq y\})\right]$

$\;$
$\;$
$\;$
"

# ╔═╡ 5f65b310-d08a-4e7c-879d-f4ebdf9255ec
#------------------------------------------------------------------------------------
# lebesgueIntegral3 only integrates *positive* functions
# - integration method is summation of all rectangles with base x and
#      function f(x) value within horizontal slab y + Δy > f(x) ≥ y
# - *non*recursive function earchForMax using *while*
#------------------------------------------------------------------------------------
function lebesgueIntegral3(f::Function; a=0.0, b=1.0, y=0.0, Δx=0.01)
	#--------------------------------------------------------------------------------
	Δy = Δx
	add_Δx(x) = x + Δx
	add_Δy(y) = y + Δy	#--------------------------------------------------------------------------------
	function μ(y)
		let μΔx(x) = (add_Δy(y) > f(x) ≥ y) ? Δx : 0
			sum(μΔx(x) for x in a:Δx:b)
		end # let
	end # function μ
	#--------------------------------------------------------------------------------
	function searchForMax(f, fmax, x, b) 
		xmax = x
		while !(add_Δx(x) > b)
			x = add_Δx(x)
			fnew = f(x)                           # new tentative maximum
			if 	fmax < fnew 
				begin
					fmax = fnew
					xmax = x
				end # begin
			end # if
			x = add_Δx(x)
		end # while
		fmax, xmax
	end # function searchForMax
	#--------------------------------------------------------------------------------
	fmax, xmax = searchForMax(f, f(a), a, b)   # the search for maximum of function f
	# vertical sum of steps f(y)*μ(f(y)) of horizontal slabs
	sum2(y -> y*μ(y), 0::Real, add_Δy::Function, fmax::Real) 
	#--------------------------------------------------------------------------------
end # function lebesgueIntegral3

# ╔═╡ 2635fd73-0552-499d-8be4-9a9323de7213
lebesgueIntegral3(x->x^2-1, b=1.0, Δx=1.0E-3) 

# ╔═╡ 6f87f654-8310-4e71-875f-a63ac41d0cd0
lebesgueIntegral3(unitPyramid, Δx=1.0E-4)             # ==> 1/4 --> :)

# ╔═╡ 15fefcd0-fc12-4c48-95f3-71a595aa6b47
abs(1/4-lebesgueIntegral3(unitPyramid, Δx=1.0E-4))

# ╔═╡ 52554fbf-327c-4ef9-a8ec-7438f930651c
areaPyramid = 0.24999176180012855
# areaPyramid = lebesgueIntegral3(unitPyramid, Δx=1.0E-5)   # ==> 0.24999176180012855 ≈ 1/4 --> :)

# ╔═╡ 413b45a7-fbeb-4176-baff-81ec3b68549e
abs(1/4 - areaPyramid)

# ╔═╡ c8bb72b1-dd6d-48e4-9664-42b96da540fa
let x1 = +1.0
	x2 = -1.0
	y1 = gaussianDensity(+1.0)
	y2 = gaussianDensity(-1.0)
	plot(gaussianDensity, -1.0, 1.0, size=(525, 400), xlim=(-3.0, 3.0), ylim=(0, 0.45), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title=L"$p(-1.0 < X < +1.0)$ for $X \sim N(X\;\;| μ=0.0, σ=1.0)$", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"$f(X)$")
	plot!(gaussianDensity, -3.0, -1.0,  line=:darkblue, label=L"$f(X)$")
	plot!(gaussianDensity, 1.0, 3.0,  line=:darkblue, label=L"$f(X)$")
	plot!([x1, x1], [0, y1], color=:red, label=L"$x=+1.0=\sigma$")
	plot!([x2, x2], [0, y2], color=:red, label=L"$x=-1.0=-\sigma$")
end # let

# ╔═╡ fdcb6e7f-2ca7-48bf-8ecc-2e06a2892ad5
lebesgueIntegral3(gaussianDensity, a=-1, Δx=1.0E-2)

# ╔═╡ 7312e9d7-df2b-491b-99a0-1429c3f6a9db
lebesgueIntegral3(gaussianDensity, a=-1, Δx=1.0E-3)

# ╔═╡ 30753208-9392-405e-83bd-8bb7c7cb2cc8
lebesgueIntegral3(gaussianDensity, a=-1.0, Δx=1.0E-4)


# ╔═╡ 20c24717-7b58-4f0c-9d97-236460ee6442
md"
---
###### Mixed Gaussians
In empirical research *mixed* Gaussians are rather seldom used to describe real data [Schilling, et al. (2002)](http://dbsi.org/dist/00031300265.pdf). In contrast to their role in data analysis mixed Gaussians are rather often used to represent prior beliefs in *Bayesian* modeling (Wikipedia, [*Mixture Model*](https://en.wikipedia.org/wiki/Mixture_model)).
"

# ╔═╡ ebdd80ce-c7b5-4e3a-92df-0b9424371805
function mixedGaussianDensity(x; μ1=0.0, μ2=0.0, σ1=1.0, σ2=1.0, w1=0.5, w2=0.5)
	w1*gaussianDensity(x, μ=μ1, σ=σ1) + w2*gaussianDensity(x, μ=μ2, σ=σ2)
end # function mixedGaussianDensity

# ╔═╡ bf8476db-30aa-4c86-a9d7-28f1e7c81663
mixedGaussianDensity(-3.0, μ1=-3.0, μ2=+3.0)

# ╔═╡ eddd8561-c278-4c00-9609-26e82c113f27
mixedGaussianDensity(+3.0, μ1=-3.0, μ2=+3.0)

# ╔═╡ 2fd2285a-4695-4802-8125-1b81e007a0da
mixedDensity(x) = mixedGaussianDensity(x, μ1=-3.0, μ2=+3.0, σ1=0.5, σ2=2.0)

# ╔═╡ 3e612d04-9ef1-4d2d-af38-70c2d4ec6fb7
begin 
	#--------------------------------------------------------------------------------
	# plot of mixed density
	plot(mixedDensity, -6.0, 10, size=(700, 500), xlim=(-6.0, 10.0), ylim=(-0.05, 0.45), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title="1st Lebesgue Integration (= Summation of Horizontal Slabs)", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"$f(X)$")
	plot!(x->0.00, -5, 10, line=:darkblue, lw=1,label="")      # base line
	#--------------------------------------------------------------------------------
	plot!(x->0.08, -5.5, 9, line=:darkblue, lw=1,label="")     # 1st horizontal line
	plot!(x->0.12, -5.5, 9, line=:darkblue, lw=1,label="")     # 1st horizontal line
	plot!([(-5.0,.08),(-5.0,.12)],line=:red,arrow=:both, label="") # 1st bi arrow
	plot!([(-5.0,.00),(-5.0,.08)],line=:red,arrow=:both, label="") # 2nd bi arrow
	annotate!(-5.5, 0.096, L"\Delta_y")
	annotate!(-5.5, 0.045, L"y")
	#--------------------------------------------------------------------------------
	plot!([(-3.75,.08),(-3.75,.12)],line=:red,lw=4,label="")     # 1st vertical line
	plot!([(-3.76,.12),(-3.76,.00)],line=:red,ls=:dash,label="") # 2nd vertical line
	plot!([(-2.24,.12),(-2.24,.00)],line=:red,ls=:dash,label="") # 3rd vertical line
	#--------------------------------------------------------------------------------
	# rectangles
	rect1 = Shape([(-3.75,.08),(-2.24,.08),(-2.24,.12),(-3.75,.12),(-3.75,0.08)])
	plot!(rect1, fillcolor = :pink)
	#--------------------------------------------------------------------------------
	annotate!(0.2,-0.03, L"\{x|(y + \Delta y) \geq f(x)\}", 10)
	plot!([(-3.10,-.03),(-3.10,-.00)],line=:red,arrow=:head, label="") # 1st up arrow
	plot!(x->.00,-3.75,-2.24,line=:red, lw=3,label="")          # 2nd horizontal line
	plot!(x->-.03,-3.10,-1.7,line=:red, label="")               # 5th horizontal line
	#--------------------------------------------------------------------------------
	plot!(mixedDensity, -6.0, 10, line=:darkblue, label="")
end # begin

# ╔═╡ df7f86f6-025b-4134-b205-ac4a078aae23
begin 
	#--------------------------------------------------------------------------------
	# plot of mixed density
	plot(mixedDensity, -6.0, 10, size=(700, 500), xlim=(-6.0, 10.0), ylim=(-0.05, 0.45), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title="2nd Lebesgue Integration (= Summation of Related Steps)", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"$f(X)$")
	plot!(x->0.00, -5, 10, line=:darkblue, lw=1,label="")      # base line
	#--------------------------------------------------------------------------------
	plot!(x->0.08, -5.5, 9, line=:darkblue, lw=1,label="")     # 1st horizontal line
	plot!(x->0.12, -5.5, 9, line=:darkblue, lw=1,label="")     # 1st horizontal line
	plot!([(-5.0,.08),(-5.0,.12)],line=:red,arrow=:both, label="") # 1st bi arrow
	plot!([(-5.0,.00),(-5.0,.08)],line=:red,arrow=:both, label="") # 2nd bi arrow
	annotate!(-5.5, 0.096, L"\Delta_y")
	annotate!(-5.5, 0.045, L"y")
	#--------------------------------------------------------------------------------
	plot!([(-3.9,.08),(-3.9,.00)],lw=3,line=:red, label="")     # 1st vertical line	
	plot!([(-3.75,.12),(-3.75,.00)],line=:red,ls=:dash,label="")# 2nd vertical line
	plot!([(-2.25,.12),(-2.25,.00)],line=:red,ls=:dash,label="")# 3rd vertical line
	plot!([(-2.1,.08),(-2.1,.00)],lw=3,line=:red, label="")     # 4th vertical line
	plot!([( 1.7,.08),( 1.7,.00)],lw=3,line=:red, label="")     # 5th vertical line
	plot!([( 1.7,.12),( 1.7,.00)],line=:red,ls=:dash,label="")  # 6th vertical line
	plot!([( 4.3,.08),( 4.3,.00)],lw=3,line=:red, label="")     # 7th vertical line
	plot!([( 4.3,.12),( 4.3,.00)],line=:red,ls=:dash,label="")  # 8th vertical line
	#--------------------------------------------------------------------------------
	# rectangles
	plot!(x->.08,-3.9,-3.75,line=:darkblue,fill=(0,:pink),framestyle=:semi, label="")
	plot!(x->0.00, -3.9,-3.75,line=:red,lw=3,label="")          # 2nd horizontal line
	plot!(x->.08,-2.25,-2.1,line=:darkblue,fill=(0,:pink),framestyle=:semi, label="")
	plot!(x->0.00,-2.25,-2.10,line=:red,lw=3,label="")          # 3rd horizontal line
	plot!(x->.08, 1.7, 4.3,line=:darkblue,fill=(0,:pink),framestyle=:semi, label="")
	plot!(x->0.0, 1.7, 4.3,line=:red,lw=3,label="")             # 4th horizontal line
	#--------------------------------------------------------------------------------
	annotate!(0.40,-0.03, L"\{x|(y + \Delta y) > f(x)\; \geq y\}", 10)
	plot!([(-3.84,-.03),(-3.84,-.00)],line=:red,arrow=:head, label="") # 1st up arrow
	plot!([(-2.15,-.03),(-2.15,-.00)],line=:red,arrow=:head, label="") # 2nd up arrow
	plot!([( 3.00,-.03),( 3.00,-.00)],line=:red,arrow=:head, label="") # 3rd up arrow
	plot!(x->-.03,-3.84,-1.90,line=:red, label="")              # 5th horizontal line
	plot!(x->-.03, 2.55, 3.00,line=:red, label="")              # 6th horizontal line
	#--------------------------------------------------------------------------------
end # begin

# ╔═╡ 3ef10af4-d6c0-4979-814b-c3674746e8ad
mixedDensity(-3.5)

# ╔═╡ c7e33dd3-3e37-44b3-a154-4fc2caec15de
let x1 = -4.0
	x2 = +6.5
	plot(mixedDensity, -4.0, 6.5, size=(700, 500), xlim=(-6.0, 10.0), ylim=(0, 0.45), line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title="Density of Mixed Gaussians", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"$f(X)$")
	#-------------------------------------------------------------------------------
	plot!(mixedDensity, -6.0, -4.0,  line=:darkblue, label=L"$f(X)$")
	plot!(mixedDensity, 6.5, 10.0,  line=:darkblue, label=L"$f(X)$")
	plot!([x1, x1], [0, mixedDensity(x1)], seriestype=:line, color=:red,  label=L"$x_1=-4.0$")
	plot!([x2, x2], [0, mixedDensity(x2)], seriestype=:line, color=:red,  label=L"$x_2=+5.5$")
end # let

# ╔═╡ c1d70fd3-5b7f-4e88-95a1-38778629d190
lebesgueIntegral1(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-3)

# ╔═╡ 45294c5a-f803-49af-bad3-43c382429392
lebesgueIntegral2(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-3)

# ╔═╡ 989ebdf9-1a46-4e4d-a4f4-6166af8cb654
lebesgueIntegral2(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-4)

# ╔═╡ 46415e65-8ee5-4b54-9243-14eb9465a404
# ==> 0.9685319771905394 --> :) 3911 sec = 65 min 18 sec
# lebesgueIntegral2(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-5) 

# ╔═╡ 21b6e8e0-1aab-4f8f-8f47-b424756b323a
lebesgueIntegral3(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-3)

# ╔═╡ 84dddd18-4c20-4162-a07c-e7b1c7219ddc
lebesgueIntegral3(mixedDensity, a=-4.0, b=6.5, Δx=1.0E-4)

# ╔═╡ 26ba2cbd-bbac-45fe-91c3-df257cde2db0
md"
---
###### Lebesgue Integration of Stick Densities

The *Lebesgue measure* (= area) unter this *stick* density is for $0.0 < x \le 3.0$ approximately $2\cdot(f(x)\cdot 2\Delta x)$.
"

# ╔═╡ b95e2161-5543-4387-892e-b05870cadd4c
function stickDensity(x; Δx=1.0E-3) 
	((1.0-Δx) <= x <= (1.0+Δx)) ? 0.5 : 
	((2.0-Δx) <= x <= (2.0+Δx)) ? 0.5 : 0.0
end # function myStickDensity

# ╔═╡ aa84697c-b915-46b6-bba4-3fa5488429db
function trueAreaOfStick(f, x; Δx=1.0E-3)
	f(x)*4*Δx
end # function trueAreaOfStick

# ╔═╡ 1ef5978e-ab58-4d22-8a16-fcb353bf9864
trueArea = trueAreaOfStick(stickDensity, 1.0)

# ╔═╡ d7412a44-26c8-468d-b52d-e30d0ae56930
begin
	xs = [x for x in 0:1E-3:3]
	#-----------------------------------------------------------------------------
	plot(xs, stickDensity, size=(700, 500), xlim=(0.0, 3.0), ylim=(0, 0.6), 
	line=:darkblue, fill=(0, :lightblue), framestyle=:semi, title="Density of Stick Function", xlabel=L"$X$", ylabel=L"$f(X)$", label=L"$f(X)$")
	#-----------------------------------------------------------------------------
	annotate!([(1.5, 0.55, L"$\int_0^3f(x)\cdot\lambda(dx)=0.5 \cdot 4\Delta x$")])
end # begin

# ╔═╡ 03f2bef6-8b0f-476a-ad4b-f186c60e1e94
approximateArea1 = lebesgueIntegral1(stickDensity, b=3.0, Δx=1.0E-3) 

# ╔═╡ 2f92b01a-d5d7-4ece-9a0e-011220570446
error1 = abs(approximateArea1 - 0.002)             # error = 0.0019939999999999654

# ╔═╡ 0fe0dc04-91fe-4656-b128-c2d40a8be217
# stack overflow, as expected
lebesgueIntegral1(stickDensity, b=3.0, Δx=1.0E-4)        

# ╔═╡ ff7860ed-a4b1-4118-a834-daeead70c70f
approximateArea2 = 0.002009959799998782     # 1020 sec = 17 min 
# approximateArea2 = lebesgueIntegral2(stickDensity, b=3.0, Δx=1.0E-5)

# ╔═╡ 73f17ff3-41cb-4e1a-8183-c32d53db82ea
# the error is smaller than with approximateArea1
error2 = abs(approximateArea2 - 0.002)             # error2 = 9.959799998781986e-6

# ╔═╡ 9f51beb7-aea9-4c3d-b2c8-6983aad2696d
error1 > error2

# ╔═╡ 48055835-3c05-4da3-931b-8df2e8a62254
approximateArea3 = 0.0020099598000014466           # 167 sec
# approximateArea3 = lebesgueIntegral3(stickDensity, b=3.0, Δx=1.0E-5)

# ╔═╡ 74be22e7-75b8-4aa5-8603-173690af51ee
error3 = abs(approximateArea3 - 0.002)             # error3 = 9.959800001446521e-6

# ╔═╡ 1724739e-f3a4-4ad5-b361-92b7247512e6
error1 > error2 > error3

# ╔═╡ 9f9832c1-d913-433f-a727-a2531b3a416a
error1 > error2 ≈ error3

# ╔═╡ 0dcea040-b70a-4a90-b0c3-47bc1a495d67
abs(error2 - error3)  # lebesgueIntegral3 is approximately equal to lebesgueIntegral2

# ╔═╡ 9bbe0e86-02a5-40ed-9a41-899323df240e
# lebesgueIntegral4(stickDensity, b=3.0, Δx=1.0E-5) # ==> 0.0020099598000014466 --> :)

# ╔═╡ 2b1e1603-963e-4d3e-b2b9-7469e54649aa
md"
---
##### 1.3.1.3 Using the Package [QuadGK.jl](https://juliapackages.com/p/quadgk)
"

# ╔═╡ 12103e5c-658a-43fa-b904-d83e3d610a63
md"
---
###### Example 6-1 in: Stark, P.A., Introduction to Numerical Methods, 1970, ch. 6.2-6.3, p. 196

"

# ╔═╡ 12a0d4d6-e259-4711-ba11-05649af921ee
let f(x) = halfParabola(x)
	a=0.0
	b=+1.0
	I,est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ 67512229-1207-4522-bfe7-87dca4feeb02
md"
###### *Riemann* fails: $QuadGK$ generates an *incorrect* integral value of $0.0$ instead of $0.002$
"

# ╔═╡ fa6643ce-8d9a-4525-825d-8a8d915118c0
let f(x) = stickDensity(x)
	a= 0.0
	b=+3.0
	I,est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ 7c734d5f-ac3d-4bbd-99b3-68c31aae6508
md"
---
###### Standard Normal (= Gaussian) Distribution using $QuadGK$
$Prob(- \sigma \le X \le +\sigma) \text{ when } X \sim N(X|\mu = 0; \sigma=1)$

$\;$
"

# ╔═╡ 8f884ca3-790c-42fd-8bcb-8476012d4ee1
let f(x) = gaussianDensity(x)
	a=-1.0
	b=+1.0
	Integral, est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ 8c1b16be-bf24-4897-8176-beff174b9f32
let f = mixedDensity
	a=-4.0
	b=6.5
	Integral, est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ ae3b9175-9e73-458d-af3d-6d5480b2a20a
md"
---
###### Integration of function $(x^2-1)$ with respect to Lebesgue measure $\lambda(dx)$ 
(NonSICP)

Note, that Hable's result is $8$ which is an *error* (Hable, 2015, p.73). This is fixed here.

$\;$
$\;$

$\int_{[0,3]}(x^2-1)\lambda(dx)=\left(\frac{1}{3}x^3-x\right)\Bigg\vert_0^3$

$\;$
$\;$
$\;$

$=\left(\frac{27}{3}-3\right)-(0-0)=6$

$\;$
$\;$
$\;$
"

# ╔═╡ 5aaeacc4-6d26-4cc5-be26-2161db3ce784
plot(x->x^2-1, 0.0, 3.0, size=(700, 300), xlims=(0.0, 3.1), ylims=(-1.5, 9.2), line=:darkblue, fill=(0, :lightblue), title=L"$x \mapsto x^2-1$", framestyle=:semi)

# ╔═╡ 85ce27de-7e09-4f96-a408-e2be7e62dde4
sum2(x->x, 0, x->x+1, 5)

# ╔═╡ e1292842-365e-4cfd-a2f9-d95fc6516860
sum2(x->x, -5, x->x+1, 0)

# ╔═╡ a344568c-f02b-4735-ad47-32c725892827
#------------------------------------------------------------------------------------
# lebesgueIntegral4 integrates *positive* and *negative* functions
# - integration method is summation of all rectangles with base x and
#      function f(x) value within horizontal slab y + Δy > f(x) ≥ y
# - *non*recursive function earchForMax using *while*
#------------------------------------------------------------------------------------
function lebesgueIntegral4(f::Function; a=0.0, b=1.0, y=0.0, Δx=0.01)
	#--------------------------------------------------------------------------------
	Δy = Δx
	add_Δx(x) = x + Δx
	add_Δy(y) = y + Δy	
	sub_Δx(x) = x - Δx
	sub_Δy(y) = y - Δy	
	#--------------------------------------------------------------------------------
	function μ(y)
		let μPositiveΔx(x) = (add_Δy(y) > f(x) ≥ y) ? Δx : 0
		    μNegativeΔx(x) = (y > f(x) ≥ sub_Δy(y)) ? Δx : 0
			μPositive = sum(μPositiveΔx(x) for x in a:Δx:b)
			μNegative = sum(μNegativeΔx(x) for x in a:Δx:b)
		    (μPositive, μNegative)
		end # let
	end # function μ
	#--------------------------------------------------------------------------------
	function searchForMax(f, fmax, x, b) 
		xmax = x
		while !(add_Δx(x) > b)
			x = add_Δx(x)
			fnew = f(x)                           # new tentative maximum
			if 	fmax < fnew 
				begin
					fmax, xmax = fnew, x
				end # begin
			end # if
			x = add_Δx(x)
		end # while
		if fmax < 0.0
			fmax, xmax = 0.0, 0.0
		end # if
		fmax, xmax
	end # function searchForMax
	#--------------------------------------------------------------------------------
	function searchForMin(f, fmin, x, b) 
		xmin = x
		while !(add_Δx(x) > b)
			x = add_Δx(x)
			fnew = f(x)                           # new tentative mainimum
			if 	fmin > fnew 
				begin
					fmin, xmin = fnew, x
				end # begin
			end # if
			x = add_Δx(x)
		end # while
		if fmin > 0.0
			fmin, xmin = 0.0, 0.0
		end # if
		fmin, xmin
	end # function searchForMin
	#--------------------------------------------------------------------------------
	fmax, xmax = searchForMax(f, f(a), a, b)  # the search for maximum of function f
	fmin, xmin = searchForMin(f, f(a), a, b)  # the search for minimum of function f
	# vertical sum of steps y*μ(f(y))
    if fmax ≥ 0.0
		positiveIntegral = sum2(y -> y*μ(y)[1], 0::Real, add_Δy::Function, fmax::Real)
	else
		positiveIntegral = 0.0
	end # if
	if fmin < 0.0
		negativeIntegral = sum2(y -> y*μ(y)[2], fmin::Real, add_Δy::Function, 0::Real)
	else
		negativeIntegral = 0.0
	end # if
    positiveIntegral + negativeIntegral
	#--------------------------------------------------------------------------------
end # function lebesgueIntegral4

# ╔═╡ 10d4b49f-f938-486f-86f8-80c13b41b73d
lebesgueIntegral4(stickDensity, b=3.0, Δx=1.0E-4) 

# ╔═╡ d6f4ab59-0a81-404d-bebf-1e89ab674079
let f(x) = x^2-1
	a=0.0
	b=3.0
	I,est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ 3676bba9-1e6f-40c7-9fa4-4ffe4b819c81
# false ! the negative part has to be subtracted ! 
lebesgueIntegral3(x->x^2-1, b=3.0, Δx=0.1E-3)  

# ╔═╡ e04a9354-1c6e-4c49-b1a2-a6b070a8d6e4
lebesgueIntegral4(x->x^2-1, b=1.0, Δx=0.1E-3) 

# ╔═╡ 4ee84ced-a30b-4452-9408-452df4b6008c
lebesgueIntegral4(x->x^2-1, b=3.0, Δx=0.1E-3)   # ==> 5.999504280000504 ≈ 6.0

# ╔═╡ 40eb47d3-6f33-47f2-9902-b1d9df4c8277
md"
---
$\;$
$\;$

$\int_{[0,3]}(x^2-1)\lambda(dx)=\left(\frac{1}{3}x^3-x\right)\Bigg\vert_1^3$

$\;$
$\;$
$\;$

$=\left(\frac{27}{3}-3\right)-\left(\frac{1}{3}-1\right)=\left(6+\frac{2}{3}\right)=6.\overline{6}$

$\;$
$\;$
$\;$
"

# ╔═╡ 5b9cccf1-10bf-47f8-9343-a5d0404d4ad3
plot(x->x^2-1, 1.0, 3.0, size=(700, 300), xlims=(0.0, 3.1), ylims=(-1.5, 9.2), line=:darkblue, fill=(0, :lightblue), title=L"$x \mapsto x^2-1$", framestyle=:semi)

# ╔═╡ 10f51d0a-a1d6-4e6c-9a9d-f7c0579c7371
let f(x) = x^2-1
	a=1.0
	b=3.0
	I,est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ 1a2d5069-9ad4-42b0-b4b1-60e5b3f18a41
lebesgueIntegral3(x->x^2-1, a=1.0, b=3.0, Δx=1.0E-3)  

# ╔═╡ fe4b73a6-3c96-45b1-8abf-3d954538c5a1
lebesgueIntegral4(x->x^2-1, a=1.0, b=3.0, Δx=1.0E-3)  

# ╔═╡ 5a481e43-eb57-4df7-a914-e72c30a69e6a
md"
---
$\;$
$\;$

$\int_{[0,3]}(x^2-1)\lambda(dx)=\left(\frac{1}{3}x^3-x\right)\Bigg\vert_0^1$

$\;$
$\;$
$\;$

$=\left(\frac{1}{3}-1\right)-\left(0-0\right)=-\frac{2}{3}=-0.\overline{6}$

$\;$
$\;$
$\;$
"

# ╔═╡ 5df40d45-72e8-4ab8-b97f-eb29f2ae7b1b
plot(x->x^2-1, 0.0, 1.0, size=(700, 300), xlims=(0.0, 3.1), ylims=(-1.5, 9.2), line=:darkblue, fill=(0, :lightblue), title=L"$x \mapsto x^2-1$", framestyle=:semi)

# ╔═╡ a9ec7875-1522-42a5-a5e2-438fe312604c
let f(x) = x^2-1
	a=0.0
	b=1.0
	I,est = quadgk(f, a, b, rtol=1e-8)
end # let

# ╔═╡ c018d013-3a67-4a61-adcc-69d252839e04
lebesgueIntegral4(x->x^2-1, b=1.0, Δx=1.0E-3) 

# ╔═╡ 84667282-7410-4c19-b227-1a2bc8bf88ee
md"
---
##### References
- **Abelson, H., Sussman, G.J. & Sussman, J.**; [*Structure and Interpretation of Computer Programs*](https://sarabander.github.io/sicp/), Cambridge, Mass.: MIT Press, (2/e), 1996; last visit 2022/08/25
- **Hable, R.**; *Einführung in die Stochastik*, Heidelberg: Springer, 2015
- **Maas, M.D.**; [*MathecDev: Gauss-Kronrod Adaptive Quadrature with QuadGK.jl*](https://www.matecdev.com/posts/julia-numerical-integration.html#gauss-kronrod-adaptive-quadrature-with-quadgkjl); last visit 2023/09/01
- **Nazarathy, Y. & Klok, H.**; *Statistics with Julia*; Cham, Switzland: Springer, 2021
- **Schilling, M.F., Watkins, A.E. & Watkins, W.**; [*Is Human Height Bimodal ?*](http://dbsi.org/dist/00031300265.pdf);  The American Statistician, Vol.56, No.3, p.223-229
- **Schilling, R.L.**; *Measures, Integrals, and Martingales*; Cambridge, UK: Cambridge University Press, 2005
- **Schilling, R.L.**; *Maß und Integral*; Berlin: de Gruyter, 2015
- **Stark, P.A.**; *Introduction to Numerical Methods*, NY: MacMillan, 1970
- **Wikipedia**; [*Dirichlet function*](https://en.wikipedia.org/wiki/Dirichlet_function); last visit 2023/09/04
- **Wikipedia**; [*Lebesgue-Integration*](https://en.wikipedia.org/wiki/Lebesgue_integration); last visit 2023/09/03
- **Wikipedia**; [*Leibnitz'* Formula for $\pi$](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80); last visit 2023/08/31
- **Wikipedia**; [*Mixture Model*](https://en.wikipedia.org/wiki/Mixture_model); last visit 2023/09/04
- **Wikipedia**; [*Normal Distribution*](https://en.wikipedia.org/wiki/Normal_distribution); last visit: 2023/08/31
- **Wikipedia**; [*Null set*](https://en.wikipedia.org/wiki/Null_set); last visit 2023/11/06
"

# ╔═╡ 9c61b414-134f-4924-87b8-8e761e5816ed
md"
---
###### end of ch. 1.3.1.1

====================================================================================

This is a **draft** under the [Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)](https://creativecommons.org/licenses/by-nc-sa/4.0/) license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

====================================================================================
"

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deps = ["Statistics"]
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[[deps.Fontconfig_jll]]
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[[deps.FreeType2_jll]]
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[[deps.GLFW_jll]]
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[[deps.GR_jll]]
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[[deps.Gettext_jll]]
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[[deps.Glib_jll]]
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[[deps.Graphite2_jll]]
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[[deps.HarfBuzz_jll]]
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[[deps.InteractiveUtils]]
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[[deps.IrrationalConstants]]
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[[deps.JLFzf]]
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[[deps.LibCURL_jll]]
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[[deps.LibGit2]]
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[[deps.Libffi_jll]]
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[[deps.Libgcrypt_jll]]
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[[deps.Libgpg_error_jll]]
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[[deps.Libuuid_jll]]
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    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"

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[[deps.NaNMath]]
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[[deps.Ogg_jll]]
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[[deps.OpenLibm_jll]]
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[[deps.OpenSSL]]
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[[deps.OpenSSL_jll]]
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[[deps.Opus_jll]]
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[[deps.Parsers]]
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[[deps.Pixman_jll]]
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[[deps.PlotThemes]]
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[[deps.PlotUtils]]
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[[deps.Plots]]
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    ImageInTerminalExt = "ImageInTerminal"
    UnitfulExt = "Unitful"

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[[deps.PrecompileTools]]
deps = ["Preferences"]
git-tree-sha1 = "03b4c25b43cb84cee5c90aa9b5ea0a78fd848d2f"
uuid = "aea7be01-6a6a-4083-8856-8a6e6704d82a"
version = "1.2.0"

[[deps.Preferences]]
deps = ["TOML"]
git-tree-sha1 = "7eb1686b4f04b82f96ed7a4ea5890a4f0c7a09f1"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.4.0"

[[deps.Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[deps.Qt6Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "364898e8f13f7eaaceec55fd3d08680498c0aa6e"
uuid = "c0090381-4147-56d7-9ebc-da0b1113ec56"
version = "6.4.2+3"

[[deps.QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "6ec7ac8412e83d57e313393220879ede1740f9ee"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.8.2"

[[deps.REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.RecipesBase]]
deps = ["PrecompileTools"]
git-tree-sha1 = "5c3d09cc4f31f5fc6af001c250bf1278733100ff"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.4"

[[deps.RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "PrecompileTools", "RecipesBase"]
git-tree-sha1 = "45cf9fd0ca5839d06ef333c8201714e888486342"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.12"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "1.0.0"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "30449ee12237627992a99d5e30ae63e4d78cd24a"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.2.0"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.SimpleBufferStream]]
git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1"
uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7"
version = "1.1.0"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "c60ec5c62180f27efea3ba2908480f8055e17cee"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.1.1"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.9.0"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "45a7769a04a3cf80da1c1c7c60caf932e6f4c9f7"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.6.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "75ebe04c5bed70b91614d684259b661c9e6274a4"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.34.0"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+6"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "9a6ae7ed916312b41236fcef7e0af564ef934769"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.13"

[[deps.URIs]]
git-tree-sha1 = "b7a5e99f24892b6824a954199a45e9ffcc1c70f0"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.5.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "a72d22c7e13fe2de562feda8645aa134712a87ee"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.17.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "93c41695bc1c08c46c5899f4fe06d6ead504bb73"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.10.3+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "cf2c7de82431ca6f39250d2fc4aacd0daa1675c0"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.4.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─015f86b0-fa99-11eb-0a32-5bee9e7368fb
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