Probabilités Tribu

TΩ Ω0Ω
T0={A∩Ω0|A∈ T } Ω0
f: Ω →Ω0T0Ω0
T=f−1(A0)|A0∈ T 0
Ω
(Ti)i∈IΩ
T=\
i∈ITi
Ω
S P(Ω) (Ti)i∈IΩ
S
T=\
i∈ITi
S
(An)n∈N(Ω,T)
A=[
p∈N\
n≥p
An
(An)n∈NA
A0=\
p∈N[
n≥p
An
Ω (An)n∈NΩ
n6=m=⇒An∩Am=∅[
n∈N
An= Ω
A=([
n∈T
An|T∈℘(N))
AΩ
Ω
Ω
(An)n∈N
Ω
T=A⊂Ω|A¯ A
TΩ
T
{ω}ωΩ
ΩT=℘(Ω)
f: Ω →Ω0
T=A⊂Ω|A=f−1(f(A))
TΩ

P (N, ℘(N))
P ({n})−−−−−→
n→+∞0
(an)n∈N
λ∈RPN, ℘(N)
P{n, n + 1, . . .}=λan
(Ω,T,P) A, B ∈ T
d(A, B) = P(A∆B)
A∆B A B
A∆B= (A∪B)\(A∩B)
d(A, C)≤d(A, B) + d(B, C)
|P(A)−P(B)| ≤ P(A∆B)
p An
n
an
a1, a2a3
an+2 anan+1 n≥1
n
pn=λn
n!e−λλ'2
n
p∈]0 ; 1[
p1

p2
n
k
pnn
pn=a2n
n!a > 0
a
(A1, . . . , An)
i∈ {1, . . . , n}f Ai=AiAi(f A1,..., f An)
(Ai)i∈I
(An)n∈N
An
exp −
+∞
X
n=0
P(An)!
(Ω,T,P) (An)n∈N
P +∞
[
n=0
An!= 1 −lim
n→+∞
n
Y
k=0
P(Ak)
P(An)6= 1 n∈N
PS+∞
n=0 An= 1
PlnP(An)
PP(An)
Ps > 1
ζ(s) =
+∞
X
n=1
n−s
λ∈Rλn−sn∈N∗
N∗
p Ap=pN∗App
P
ζ(s) = Y
p∈P
1
1−p−s
(1/p)p∈P

Ω0= Ω ∩Ω0Ω∈ T Ω0∈ T 0
B∈ T 0B=A∩Ω0A∈ T Ω0\B=¯ A∩Ω0
¯
A∈ T BΩ0T0
(Bn)T0Bn=An∩Ω0An∈ T
+∞
[
n=0
Bn= +∞
[
n=0
An!∩Ω0
+∞
[
n=0
An∈ T
+∞
[
n=0
Bn∈ T 0
Ω = f−1(Ω0) Ω0∈ T 0
Ω∈ T
A∈ T A0∈ T 0A=f−1(A0)
¯
A=f−1A0A0∈ T 0
¯
A∈ T 0
(An)n∈N∈ T N(A0
n)n∈N∈ T 0N
∀n∈N, An=f−1(A0
n)
+∞
[
n=0
An=f−1 +∞
[
n=0
A0
n!+∞
[
n=0
A0
n∈ T 0
+∞
[
n=0
An∈ T
ΩTiT
A∈ T ATi¯ A
¯
A∈ T
(An)∈ T Ni∈I(An)∈(Ti)NSn∈NAn∈ Ti
Sn∈NAn∈ TTT
ASA∈ Tii∈I
A∈ T T S
T0S
(Ti)i∈I
T=\
i∈ITi⊂ T 0
p∈NTn≥pAn
A
ATn≥pAn
p(An)
An
A0
A0
A0=[
p∈N\
n≥p
An
An
A0An
ϕ: Ω →Nω n ∈N
ω∈An
T⊂Nϕ−1(T) = Sn∈TAnA
℘(N)ϕ

AΩ
RΩω ω0
∀A∈ A, ω ∈A⇐⇒ ω0∈A
RΩ
Ω
∀A∈ A, ω ∈A=⇒Cl(ω)⊂A
ω0/∈Cl(ω)A∈ A
(ω∈A ω0/∈A) (ω /∈A ω0∈A)
A ω ∈A ω0/∈A Aω0
Cl(ω) = \
ω0/∈Cl(ω)
Aω0∈ A
A
ω
ω0
Aω0
AA
A=[
ω∈A
Cl(ω)
A
(An)n∈N
A
[
n∈T
AnT∈℘(N)
℘(N)
¯
Ω = ∅¯ Ω Ω ∈ T
T(An)n∈NT
An
Sn∈NAn
An
An0An0
[
n∈N
An⊂\
n∈N
An⊂An0
Sn∈NAn
(An)n∈NT
T {ω}ωΩ
A {ω}ωΩ
Ω
A
A
T ⊂ A
Ω Ω
{ω} T
℘(Ω) = T
Ω = f−1(f(Ω)) Ω ∈ T
A∈ T ¯ A∈ T ¯ A=f−1f¯ A x∈f−1f¯
A
y∈¯ A f(x) = f(y)x∈A y ∈f−1(f(A)) = A
x∈¯ A f−1f¯ A⊂¯ A
(An)n∈NT
f +∞
[
n=0
An!=
+∞
[
n=0
f(An)
f−1 f +∞
[
n=0
An!!=
+∞
[
n=0
f−1(f(An)) =
+∞
[
n=0
An