corrigé

E
∅
{E} {∅}
E={1,2,3,4} {∅, E, {1,2},{3,4},{2,3},{1,4}}
{1,2}∪{2,3}
E E =N
a, b a < b R
[a, b] ]a, b] ] − ∞, a]{a}Z Q
[a, b] ] − ∞, a]{a}Z
Q
]a, b] = [a, b]∩]a, +∞[
Ω = {0,1}N0 1 i≥1
ε∈ {0,1}
ci,ε ={ω= (ω0, ω1, . . .)∈Ω : ωi=ε}.
FΩCi,ε

ΩF
A={1},
B={1},
Ca,b,N =(ω= (ωn)n≥0:a≤
N−1
X
i=0
ωi≤b), N ∈N, a, b ∈Ra < b,
Dp=(ω= (ωn)n≥0: lim
n→∞
1
n
n−1
X
i=0
ωi=p), p ∈[0,1],
E=(ω= (ωn)n≥0: lim
n→∞
1
n
n−1
X
i=0
ωi).
E=Sp∈[0,1] DpE
Dp
A=\
n∈N[
i≥n
ci,1.
ci,1F
FSi≥nci,1n∈N
Fci,ε
B=\
n∈N
(ci,1∩ci+1,1)c.
Ca,b,N =[
I⊂{0,...,N−1}
a≤#I≤b\
i∈I
ci,1\
j∈{1,...,N−1}\I
cj,0.
I I P({0, . . . , N−1})
Ca,b,N F
Dp=\
k≥1[
n≥0\ i≥n
Ci(p−1
k),i(p+1 k),i.
E
E=\
k≥1[
n≥0[
a∈Q\
i≥n
Ci(a−1
k),i(a+1 k),i.

C2
4= 6 6 ×8 = 48
3
30 =1 10
30 ×29
6×2 + 24 ×3 = 80
80
30×29 =8 87 <1 10
n≥1
n
k≥1k
n
p
1−p p = 1/2
p
n
n−1 (1 −p)n−1p
k≥1k n
k−1n−1
n Ck−1
n−1
k−1n−1
(1 −p)n−kpk
k n Ck−1
n−1(1 −p)n−kpk
R

ΩL P(Ω) λΩ∈L
A, B ∈LA⊂B B \A∈L
(An)n≥0LSn≥0AnL
S P(Ω) λ
SλS
λS
I P(Ω) Iπ n ≥1
A1, . . . , An∈IA1∩. . . ∩An∈I
λ π
F P(Ω) λ π
FA1, . . . , An∈F
A1∪. . . ∪An∈F F
SπLλS
L S
L
A∈S
L1={B∈L:A∩B∈L}.
L1λS L1=L
A∈L
L2={B∈L:A∩B∈L}.
L2λS L2=L
Sπ λ
S S
(Ω,F)P1P2
(Ω,F)SπF F =σ(S)
P1(A) = P2(A)A∈SP1=P2
E={A∈F:P1(A) = P2(A)}.
]− ∞, a]a∈Rπ
R(R,BR)

λ
λ
λS
λ
SP(Ω) λSλS
λ
∅
λ∅= Ω \Ω
π
n= 2
π λ
A1A2FA1∪A2= (Ac
1∩Ac 2)c
AnF
n≥2Bn:= A1∪· · · ∪ An∈F F
λSn≥1AnBn
F F
SλS
L
S
Ω∈L1A∈S
A∩Ω = A∈S⊂L
B1B2L1B2⊂B1L1
A∩B1A∩B2L L λ
A∩(B1\B2)=(A∩B1)\(A∩B2)∈L.
B1\B2∈L1
(Bn)n≥1L1A∩Bn
L
A∩([
n≥1
Bn) = [
n≥1
(A∩Bn)∈L.
Sn≥1Bn∈L1
L1λ A ∈S S π
L1S L λS L ⊂L1
L1L1⊂L L1=L