21.4. FUBINI’S THEOREM FOR BOCHNER INTEGRALS 667

is λ measurable. In terms of nonnegative functions which are F ×S measurable,

s1 → f (s1,s2) is µ measurable,s2 → f (s1,s2) is λ measurable,

s1 →∫

Ω2

f (s1,s2)dλ is µ measurable,

s2 →∫

Ω1

f (s1,s2)dµ is λ measurable,

and the conclusion of Fubini’s theorem holds.∫Ω1×Ω2

f d (µ×λ ) =∫

Ω1

∫Ω2

f (s1,s2)dλdµ

=∫

Ω2

∫Ω1

f (s1,s2)dµdλ .

The following theorem is the version of Fubini’s theorem valid for Bochner integrablefunctions.

Theorem 21.4.1 Let f : Ω1×Ω2→ X be strongly measurable with respect to µ ×λ andsuppose ∫

Ω1×Ω2

|| f (s1,s2)||d (µ×λ )< ∞. (21.4.17)

Then there exist a set of µ measure zero, N and a set of λ measure zero, M such that thefollowing formula holds with all integrals making sense.∫

Ω1×Ω2

f (s1,s2)d (µ×λ ) =∫

Ω1

∫Ω2

f (s1,s2)XN (s1)dλdµ

=∫

Ω2

∫Ω1

f (s1,s2)XM (s2)dµdλ .

Proof: First note that from 21.4.17 and the usual Fubini theorem for nonnegative valuedfunctions, ∫

Ω1×Ω2

|| f (s1,s2)||d (µ×λ ) =∫

Ω1

∫Ω2

|| f (s1,s2)||dλdµ

and so ∫Ω2

∥ f (s1,s2)∥dλ < ∞ (21.4.18)

for µ a.e. s1. Say for all s1 /∈ N where µ (N) = 0.Let φ ∈ X ′. Then φ ◦ f is F ×S measurable and∫

Ω1×Ω2

|φ ◦ f (s1,s2)|d (µ×λ )

≤∫

Ω1×Ω2

∥φ∥∥ f (s1,s2)∥d (µ×λ )< ∞