Fubini's theorem for L^1 functions on a sigma-finite product

By [L2], the function $\mathrm{\mid }f\mathrm{\mid }$ is nonnegative measurable, and [L1] gives ${\int }_X\left({\int }_Y\mathrm{\mid }{f}_x\mathrm{\mid }d\nu \right)d\mu ={\int }_{X\times Y}\mathrm{\mid }f\mathrm{\mid }d(\mu \times \nu )<\mathrm{\infty }\mathrm{.}$ Therefore ${\int }_Y\mathrm{\mid }{f}_x\mathrm{\mid }d\nu <\mathrm{\infty }$ for $\mu$-almost every $x$, so $f_x\in L^1(\nu )$ for $\mu$-almost every $x$. The same argument with the variables reversed gives $f^y\in L^1(\mu )$ for $\nu$-almost every $y$ and shows that the section integrals of $\mathrm{\mid }f\mathrm{\mid }$ are integrable.

For almost every $x$ from step 1.1, [L4] gives $\mid {\int }_Y{f}_{x}d\nu \mid \le {\int }_Y\mathrm{\mid }{f}_x\mathrm{\mid }d\nu \mathrm{.}$ Define the section integral to be $0$ on the exceptional null set where $f_x\mathrm{\notin }{L}^1(\nu )$. The right-hand side is integrable over $X$ by step 1.1, so this extended section-integral function is integrable. The same convention and conclusion hold for $y\mapsto {\int }_X{f}^{y}d\mu$.

Write $f=u-v+i(p-q)$, where $u,v,p,q\ge 0$ are the positive and negative parts of the real and imaginary parts of $f$. Each of $u,v,p,q$ is integrable because $\mathrm{\mid }u\mathrm{\mid },\mathrm{\mid }v\mathrm{\mid },\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid }\le \mathrm{\mid }f\mathrm{\mid }$. Applying [L1] to these four nonnegative functions and recombining with [L3] yields the stated equality of the three integrals.