24.4. FUBINI’S THEOREM FOR BOCHNER INTEGRALS 669

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

Theorem 24.4.1 Let f : Ω1×Ω2→ X be strongly measurable with respect to µ×λ

and suppose ∫Ω1×Ω2

|| f (s1,s2)||d (µ×λ )< ∞. (24.18)

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)XNC (s1)dλdµ

=∫

Ω2

∫Ω1

f (s1,s2)XMC (s2)dµdλ .

Proof: First note that from 24.18 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 < ∞ (24.19)

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 )< ∞

and so from the usual Fubini theorem for complex valued functions,∫Ω1×Ω2

φ ◦ f (s1,s2)d (µ×λ ) =∫

Ω1

∫Ω2

φ ◦ f (s1,s2)dλdµ. (24.20)

Now also if you fix s2, it follows from the definition of strongly measurable and theproperties of product measure mentioned above that s1 → f (s1,s2) is strongly measur-able. Also, by 24.19

∫Ω2∥ f (s1,s2)∥dλ < ∞ for s1 /∈ N. Therefore, by Theorem 24.2.4

s2→ f (s1,s2)XNC (s1) is Bochner integrable. By 24.20 and 24.6∫Ω1×Ω2

φ ◦ f (s1,s2)d (µ×λ )

=∫

Ω1

∫Ω2

φ ◦ f (s1,s2)dλdµ

=∫

Ω1

∫Ω2

φ ( f (s1,s2)XNC (s1))dλdµ

=∫

Ω1

φ

(∫Ω2

f (s1,s2)XNC (s1)dλ

)dµ. (24.21)

24.4, FUBINI’S THEOREM FOR BOCHNER INTEGRALS 669The following theorem is the version of Fubini’s theorem valid for Bochner integrablefunctions.Theorem 24.4.1 Lez f 2 Q4 x Q2 > X be strongly measurable with respect to ux Aand suppose| If (51559) ||d (Ux A) <ee, (24.18)Q) xQoThen there exist a set of & measure zero, N and a set of A measure zero, M such that thefollowing formula holds with all integrals making sense.J flovmdtuxay = [Pf tloise) ye (or)ddawQ) x Qo Qy JQ= | F (51452) ye (so) dda.Q) JQ,Proof: First note that from 24.18 and the usual Fubini theorem for nonnegative valuedfunctions,[, g.lirevsaiatwxay= ff lires)ilaaaand so[ Il f (81,82) || dA <0 (24.19)Qofor Ll a.e. s;. Say for all s; ¢ N where u (N) = 0.Let @ € X’. Then 0 f is ¥ x. measurable and| |9 of (s1,82)|d(u x A)Q) xQ2S | IO If (s1,82)|]d (Ux A) <0Q) x Qoand so from the usual Fubini theorem for complex valued functions,J, geforn)atuxay= ff oo ftsisn)ddau. 24.20)Q) x QoQy JQNow also if you fix s2, it follows from the definition of strongly measurable and theproperties of product measure mentioned above that s; > f(s1,s2) is strongly measur-able. Also, by 24.19 fo, || (81,52)||dA < ce for s; ¢ N. Therefore, by Theorem 24.2.482 — f (81,82) Lye (s1) is Bochner integrable. By 24.20 and 24.6| bof (si,s2)d(u xd)Q) x Qo= [ of (s1,82)dAduQ, Ja,=f [oF 182) Bye (sn)= Le ([,to19) Me (x) da) dl. (24.21)