Lemme de Borel-Cantelli et modes de convergence

(Ω,F,P)
(An)n≥1
(An)n≥10
(An)n≥1L20
(An)n≥10
An0
P(An>1
2) = P(An) 0 P(An)
0ε > 0P(An> ε)≤P(An) 0
limn→∞ P(An)=0
E[An] = P(An) limn→∞ P(An) = 0
ω∈Ω ( An(ω))n≥10
0ω
lim inf Ac
nlim sup An
P(lim sup An)=0
Ω (Xn)n≥1
Ω
Ω
A=[
a∈R[
b∈R\
n≥1
{ω∈Ω : a≤Xn(ω)≤b},
B=[
N≥1\ n≥N\
m≥n
{ω∈Ω : Xn(ω)−Xm(ω)≥0},
C=[
`∈R+\ k≥1[
N≥1\ n≥Nω∈Ω : |Xn(ω)−`| ≤ 1
k,
D=[
k≥1\
N≥1[ n≥N[ m≥Nω∈Ω : |Xn(ω)−Xm(ω)|>1
k.

Ω
ω∈Ω (Xn(ω))n≥1
E
F
G+∞
H
A, B, C, D
ω∈Ω (Xn(ω))n≥1
A
B
C
D
R
D ω ∈Ω (Xn(ω))n≥1
D=\
`∈R[ k≥1\
N≥1[ n≥Nω∈Ω : |Xn(ω)−`|>1
k.
(Xn)n≥1
D
C
(Ω,F,P) (Xn)n≥1
C
C C

E=\
n≥1
{ω∈Ω : Xn(ω)≥0},
F=\
M≥1[
n≥1
{ω∈Ω : Xn(ω)≥M},
G=\
M≥1[ N≥1\ n≥N
{ω∈Ω : Xn(ω)≥M},
H=\
`∈R+[ k≥1\
N≥1[ n≥Nω∈Ω : |Xn(ω)−`|>1
k
= [
k≥1\
N≥1[ n≥N[ m≥Nω∈Ω : |Xn(ω)−Xm(ω)|>1
k!
∪ [
k≥1\
N≥1[ n≥Nω∈Ω : Xn(ω)≤ − 1
k!.
Ω (An)n≥1Ω
Ω
[
N≥1\ n≥N
An,\
N≥1[ n≥N
An,[
n≥1
An,\
n≥1
An,Ω,∅.
Ω
Ω Ω
ω∈Ω
\
n≥1
AnAn
[
N≥1\ n≥N
AnAn
\
N≥1[ n≥N
AnAn
[
n≥1
AnAn
Ω
∅⊂\
n≥1
An⊂[
N≥1\ n≥N
An⊂\
N≥1[ n≥N
An⊂[
n≥1
An⊂Ω.

Ω = {1,2,3,4,5}A1={1}A2={1,2,3,4}
A3={1,2}A4={1,2,3}A2n+1 ={1,2}A2n+2 ={1,2,3}n≥2
\
n≥1
An={1},[
N≥1\ n≥N
An={1,2},\
N≥1[ n≥N
An={1,2,3},[
n≥1
An={1,2,3,4}
(an)
n→∞
an= lim
n→∞ sup
k≥n
akn→∞ an= lim
n→∞ inf
k≥nak
(An)
An=\
n≥1[
k≥n
AkAn=[
n≥1\
k≥n
Ak
lim sup An={ω∈Ω : {n:ω∈An} }
(Ω,F,P) (An)n≥1
X
n≥1
P(An)<+∞.
P(lim sup An)=0
(An)n≥1
X
n≥1
P(An) = +∞.
P(lim sup An)=1
x1 + x≤ex
n, m 1≤m≤n
P n
\
k=m
Ac
k!≤exp −
n
X
k=m
P(Ak)!.
m≥1P ∞
\
k=m
Ac
k!= 0

[
n≥k
Ank
k
P [
n≥k
An!&P \
k≥1[ n≥k
An!=P(lim sup An).
P [
n≥k
An!≤X
n≥k
P(An).
P(An)
0k
P(lim sup An)
x= 0
x=−P(Ak)k≥1
P(Ac
k) = 1 −P(Ak)≤e−P(Ak).
k=m, . . . , n
P n
\
k=m
Ac
k!=
n
Y
k=m
P(Ac
k)≤exp −
n
X
k=m
P(Ak)!.
n
n
n
\
k=m
Ac
k
P(T∞
k=mAc
k) = 0
P ∞
\
k=m
Ac
k!= lim
n→∞
P n
\
k=m
Ac
k!= 0.
m≥1
∞
\
k=m
Ac
k
lim sup(An)