Intégrale de Lebesgue

f: [a, b]−→ R
N−1
i=0
f(ξi) (xi+1 −xi), a =x0≤x1≤x2≤ · · · ≤ xN=b, ξi∈[xi, xi+1]
h= sup
0≤i≤N−1
|xi+1 −xi|0
f[a, b]
[a, b]
S
y=f(x)
S f

(S)'
N−1
i=0
yi(Ai),
(Ai)
Ai=f−1(]yi, yi+1]) = {x∈[a, b], yi< f(x)≤yi+1}
0 = y0< y1<· · · < yN=[a,b]f
1
y
y
y
y
y
y
y
y
2
3
4
5
6
7
8
y0
1
2
3
4
5
6
7
A =
A =
A =
A =
A =
0
A = Vide
A = Vide
A = Vide
XR
P(X)X

XTX
T ⊂ P(X)
∅XT
A∈ T X\A∈ T
(An)n∈NXT
n∈N
AnT
TX
X
T0={∅, X}
T∞=P(X)
A
XTX
Tµ:T → [0,+∞]
µ(∅) = 0
(An)n∈NT
σ
µ
n∈N
An=
n∈N
µ(An)
XT
XTµ
(X, T, µ)
(X, T, µ)An
TA0⊂A1⊂A2⊂. . . B =n∈NAn
B∈ T
µ(B) = lim
n→+∞µ(An)

(X, T, µ)A∈ T
µ(A) = 0
Rd
X=RdB(Rd)
RdB(Rd)
Rd
R Rda≤b
(a, b) [a, b] ]a, b[ [a, b[ ]a, b]
µ:B(R)→[0,+∞]
R(a, b)a≤b µ((a, b)) = b−a
µ:B(Rd)→[0,+∞]
RdP= Πd
i=1(ai, bi)
µ(P) = Πd
i=1(bi−ai)
(bi−
ai)R2
R3
Rd
RdB(Rd)
f:Rd→R
∀α∈R, f−1(]α, +∞[) ∈ B(Rd)
f
B(R)B(Rd)

f g
|f|f+g fg max(f, g)min(f, g)
f/g
fninfn∈Nfn(x)
supn∈Nfn(x)fn
f f
A∈ B(Rd)
χA
χA(x) = 1 x∈A, 0
χA
e
e(x) =
N
i=1
αiχAi(x)
AiRd
Rd
X
Rdαi≥0i= 1 N Ai⊂X
Rd
e(x) =
N
i=1
αiχAi(x)
e X X e(x)dx
R+∪ {+∞}
X
e(x)dx =
N
i=1
αiµ(Ai).