Fubini's theorem for Bochner Integral

Question

I've just been (as of two days ago) introduced to the Bochner integral, and I've read that Fubini's theorem holds for it, but I haven't been able to find its version for the said integral. So here's my question:

Let $(X,\Sigma,\mu),(Y,\Omega,\upsilon)$ be $\sigma$-finite measure spaces, $E$ a Banach space and $f:X\times Y\rightarrow E$ a Bochner integrable function. I've seen (through here, page 11, theorem 1.19) that a closed linear operator (as I think is the case for the integral of Bochner integrable functions) commutes with the Bochner integral. Can I then affirm that $$ \int_{X}\left(\int_{Y}f\ d\upsilon\right)d\mu=\int_{Y}\left(\int_{X}f\ d\mu\right)d\upsilon$$ ? I apologize in advance if anything is wrong or nonsense. As I said, I've just come in contact with this subject.

Answer 1

I think that your idea does not work immediately. Indeed, if $f \in L^1(X; F)$, you can take a bounded linear operator $A \colon F \to G$ and have $$ \int_X A f(x) \, \mathrm{d}x = A \int_X f(x) \, \mathrm{d}x. $$ This does not yield your desired formula in case $f \in L^1(X \times Y; E)$, since the integral over $Y$ is not a linear operator on the space $X$.

However, the space $L^1(X \times Y; E)$ is isometrically isomorphic to $L^1(X; L^1(Y;E))$. Let $I \colon L^1(Y;E) \to E$ be the Bochner integral. Then, $$ \int_X \int_Y f \, \mathrm{d}y \, \mathrm dx = \int_X I f \,\mathrm dx = I \int_X f \,\mathrm dx = \int_Y\int_X f \,\mathrm{d}x\,\mathrm{d}y $$ for all $f \in L^1(X; L^1(Y;E))$.

One should be aware that the proof of the mentioned identification of spaces might already require some Fubini-type argument (I do not remember exactly...)

Answer 2

It follows from the usual Fubini theorem. Let $p\in E^*$ be a bounded linear functional. Then, by the usual Fubini theorem and the fact that you mentioned, $$p\int_{X}\int_{Y}f\ \mathrm d\upsilon\ \mathrm d\mu= \int_{X}p \int_{Y}f\ \mathrm d\upsilon\ \mathrm d\mu=\int_{X}\int_{Y}pf\ \mathrm d\upsilon\ \mathrm d\mu$$

$$=\int_{Y}\int_{X}p f\ \mathrm d\mu\ \mathrm d\upsilon=\int_{Y}p\int_{X}f\ \mathrm d\mu\ \mathrm d\upsilon=p\int_{Y}\int_{X}f\ \mathrm d\mu\ \mathrm d\upsilon$$

Since bounded linear functionals separate the elements of $E$ by the Hahn-Banach theorem, we get

$$\int_{X}\int_{Y}f\ \mathrm d\upsilon\ \mathrm d\mu=\int_{Y}\int_{X}f\ \mathrm d\mu\ \mathrm d\upsilon.$$