Differentiation Under the Integral Sign

Theorem (Differentiation under the integral). Suppose (Ω, F, µ) is a measure

space and I ⊂ R is an open interval. Let f : I × Ω → R be a function

with the property that f(t, ω) is differentiable wrt t for all ω

and integrable wrt ω for every t. Suppose that there is an integrable function g on Ω

so that ∂f(t, ω) ∂t ≤ g(ω)

for all t ∈ I. Then the function ω 7→ ∂f(t,ω) ∂t is integrable, the function

F : t 7→ Z Ω f(t, ω) dµ(ω)

is differentiable and d dt Z Ω