TD5. Fonctions de répartition

e
a≤b∈RI= [a, b]
FI:R→R, x 7→ 
0x≤a,
x−a x ∈I,
b−a x ≥b.
FI
λIλI(] − ∞, x]) x∈R
A∈ BRλI(A) = λ(A∩I)λ
RλI(I)
λI
I= [0,1] ϕ:I→Cx7→ 2iπx
C=ϕ(I)ϕ([x, y]) x≤y∈I
λI([x, y]) ϕ([x, y])
AR2µ(A) = λI(ϕ−1(A∩C))
(R2,BR2)
f x ≤y∈R
µ[x, y[=Zy
x
f(t)dt ,
F:x7→ Rx
0f(t)dt R
µ(R,BR)µ(R)
µ
fRI
f µ([x, y[) = µ([|y|,|x|[) x≤y≤0
f(t) = 1[−1,1](t)√1−t2t∈Rµ([0,1]) t= cos u
µ(R)µ
F
F:R→R, x 7→
x x ≤0,
2x0≤x≤1/2,
2x+ 1 1/2< x ≤1,
x+ 2 1 ≤x≤2,
x+ 4 x > 2

F µ
(R,BR)
y F µ({y})
µdµ
µc=µ−µdµ
R
λ A ⊂R
µ(A) = X
n∈A∩N
λn
n!
−λ.
µ(R,P(R)) µ
λ
R
Q={qn, n ≥1}
µ=X
n≥1
1
n2δqn,
δxx
µ(R,BR)µ({x}) = 0
x∈R\Q
µ
µ
TR
T(x) = 
x0≤x < 1,
x−1 1 ≤x≤2,
0
µ σ F
T6
T−1(I)I1
2,5 6 3 4,7 3 −8,−1 4
I T −1(I)
T
Tµµ T Tµ(B) = µT−1(B)
TR
µ=δ4/3
µ[0,2[
µ3
4,7 3