Compléments de théorie de la mesure Fichier

f E F A
E f
f(A) = {f(x):x∈A}={y∈F:∃x∈A y =f(x)}.
(Ai)i∈I E
f(S
i∈I
Ai) = S
i∈I
f(Ai)
f(T
i∈I
Ai)⊂T
i∈I
f(Ai)
f f(T
i∈I
Ai) = T
i∈I
f(Ai)
f E F
B F f−1(B) = {x∈E f(x)∈B}
x∈f−1(B)⇐⇒ f(x)∈B.

g F G A G
(g◦f)−1(A) = f−1[g−1(A)].
f−1(F) = E
(Ai)i∈I F A B F
f−1(Ac) = [f−1(A)]c
f−1(S
i∈I
Ai) = S
i∈I
f−1(Ai)
f−1(T
i∈I
Ai) = T
i∈I
f−1(Ai)
f−1(A−B) = f−1(A)−f−1(B)
f−1(A∆B) = f−1(A)∆f−1(B)
x∈f−1(Ac)⇔f(x)∈Ac⇔f(x)/∈A⇔x /∈f−1(A)⇔x∈[f−1(A)]c
x∈f−1(S
i∈I
Ai)⇔f(x)∈S
i∈I
Ai⇔ ∃i∈ I f(x)∈Ai⇔ ∃i∈ I x∈f−1(Ai)
⇔x∈S
i∈I
f−1(Ai)
(E, E, µ)µ σ [0,+∞] (F, F)
ϕ(E, E) (F, F)
µ ϕ µϕ(F, F)
∀A∈ F µϕ(A) = µ[ϕ−1(A)].

µϕ(An)n≥1
FA=S
n≥1
An
µϕ(A) = µ[ϕ−1(A)] = µ[ϕ−1([
n≥1
An)] = µ[[
n≥1
ϕ−1(An)] = X
n≥1
µ[ϕ−1(An)] = X
n≥1
µϕ(An),
ϕ−1(An)i j
ϕ−1(Ai)∩ϕ−1(Aj) = ϕ−1(Ai∩Aj) = ϕ−1(∅) = ∅.
µ µϕµϕ(F) = µ[ϕ−1(F)] = µ(E)
(G, G)ψ(F, F)
(G, G) (µϕ)ψ=µψ◦ϕAG
µψ◦ϕ(A) = µ[(ψ◦ϕ)−1(A)] = µ[ϕ−1(ψ−1(A))] = µϕ[ψ−1(A)] = (µϕ)ψ(A).
h(F, F) [0,+∞]h◦ϕ
Rhdµϕ=Rh◦ϕdµ
h(F, F)
[0,+∞[h=
n
P
i=1
ai1Aiai∈[0,+∞[Ai∈ F
Zhdµϕ=Z[
n
X
i=1
ai1Ai]dµϕ=
n
X
i=1 Zai1Aidµϕ=
n
X
i=1
aiZ1Aidµϕ
=
n
X
i=1
aiµϕ(Ai) =
n
X
i=1
aiµ[ϕ−1(Ai)] =
n
X
i=1
aiZ1ϕ−1(Ai)dµ =
n
X
i=1
aiZ(1Ai◦ϕ)dµ
=Z[
n
X
i=1
ai1Ai◦ϕ]dµ =Z[
n
X
i=1
ai1Ai]◦ϕdµ =Zh◦ϕdµ.
h(hn)n
(F, F) [0,+∞[
h(hn◦ϕ)nh◦ϕ
Zhdµϕ= lim
n→+∞Zhndµϕ= lim
n→+∞Zhn◦ϕdµ =Zh◦ϕdµ.

h(F, F)Rµϕ
h◦ϕ µ Rhdµϕ=Rh◦ϕdµ
h µϕ
⇔R|h|dµϕ<∞
⇔R|h| ◦ ϕdµ < ∞
⇔R|h◦ϕ|dµ < ∞
⇔h◦ϕ µ
Zhdµϕ=Zh+dµϕ−Zh−dµϕ=Z(h◦ϕ)+dµ −Z(h◦ϕ)−dµ =Zh◦ϕdµ.
(E, E, µ)h(E, E) [0,+∞]
E −→ [0,+∞]A−→ RAhdµ =R1Ahdµ
h µ hµ
hµ (An)n≥1
EA=S
n≥1
An
hµ(A) = Zh1Adµ =Zh1∪
n≥1Adµ =Zh(X
n≥1
1An)dµ
=ZX
n≥1
(h1An)dµ =X
n≥1Z(h1An)dµ =X n≥1
hµ(An).
A∈ E µ(A)=0⇒hµ(A)=0 hµ
µ hµ µ
h0(E, E) [0,+∞]
h=h0µ µ(h6=h0) = 0 hµ =h0µ
hµ =h0µ µ σ h =h0µ
E={e}µ({e}) = +∞h(e) = 1 h0(e) = 2 +∞hµ =h0µ=µ

h(E, E) [0,+∞]ν=hµ
f(E, E) [0,+∞]Rfdν =Rhfdµ
f(E, E)Rf ν
hf µ Rfdν =Rhfdµ
f(E, E) [0,+∞[f=
n
P
i=1
ai1Aiai∈[0,+∞[Ai∈ E
Zfdν =
n
X
i=1 Zai1Aidν =
n
X
i=1
aiν(Ai) =
n
X
i=1 Zaih1Aidµ =Zh(
n
X
i=1 Zai1Ai)dµ =Zhfdµ.
f
f ν
⇔R|f|dν < ∞
⇔R|f|hdµ < ∞
⇔R|hf|dµ < ∞
⇔hf µ
h h0R[0,+∞]h=h0
m1h=h0
ϕ(E, E, µ) (F, F)h
(F, F) [0,+∞]g=h◦ϕ(gµ)ϕ=hµϕ
A∈ F
hµϕ(A) = Z1Ahdµϕ
=Z(1A◦ϕ)(h◦ϕ)dµ
=Z(1A◦ϕ)gdµ
=Z(1A◦ϕ)d(gµ)
=Z1Ad(gµ)ϕ
= (gµ)ϕ(A).