Preliminary Notation and Definitions
In the following theorems, we will consider:
• (Ω, F, µ) is a measure space and R+ := [0, ∞)
• L1 (Ω, F, µ) := {f : Ω → R : f is F -measurable and
R
|f | dµ < ∞}
R
• L1+ (Ω, F, µ) := {g : Ω → R+ : g is F -measurable and g dµ < ∞}
• N := {0, 1, 2, . . . } and N∗ := N \ {0}
• p ∈ N∗ and [[1, p]] := N ∩ [1, p]
• ∀v ∈ Rp , ∥v∥∞ := max{|vi | : i ∈ [[1, p]]}
• U is an open set of Rp
If f : U × Ω → R is a function such that:
f (·, ω) ∈ C 1 (U, R)
∀ω ∈ Ω,
Then for all i ∈ [[1, p]], we define:
∂f
: U × Ω → R by:
• ∂x
i
∂f
f (x + tei , ω) − f (x, ω)
(x, ω) := lim
t→0
∂xi
t
• ∇x f : U × Ω → Rp by:
∇x f (x, ω) :=
∂f
∂f
(x, ω), . . . ,
(x, ω)
∂x1
∂xp
Theorem A 1. Let f : U × Ω → R be a function such that:
1. ∀x ∈ U, f (x, ·) is F -measurable
2. ∀ω ∈ Ω, f (·, ω) ∈ C(U , R)
3. ∀x ∈ U, ∃r > 0, ∃g ∈ L1+ (Ω, F, µ):
V = B(x, r) ⊂ U
∀y ∈ V, ∀ω ∈ Ω, |f (y, ω)| ≤ g(ω)
and
Define F : U → R by:
Z
F (x) :=
f (x, ω) dµ(ω)
Ω
Then:
F ∈ C(U, R)
1
Theorem B 1. Let f : U × Ω → R be a function such that:
1. ∀x ∈ U, f (x, ·) ∈ L1 (Ω, F, µ)
2. ∀ω ∈ Ω, f (·, ω) ∈ C 1 (U, R)
3. ∀x ∈ U, ∃ρ > 0, ∃h ∈ L1+ (Ω, F, µ):
W = B(x, ρ) ⊂ U
and
Define F : U → R by:
∀y ∈ W, ∀ω ∈ Ω, ∥∇x f (y, ω)∥∞ ≤ h(ω)
Z
F (x) :=
f (x, ω) dµ(ω)
Ω
Then:
F ∈ C 1 (U, R)
and
∀i ∈ [[1, p]] , ∀x ∈ U,
2
∂F
(x) =
∂xi
Z
∂f
(x, ω) dµ(ω)
Ω ∂xi