Tonelli and Fubini for the completed product, with only almost-everywhere section measurability

In the nonnegative branch, [L1] gives a product-measurable extended-real function $g$ such that $f=g$ almost everywhere for $\overline{\mu \times \nu }$. Let $N:=\{(x,y):f(x,y)\mathrm{\ne }g(x,y)\}$, so $\overline{\mu \times \nu }(N)=0$. By the completion definition, choose a product-measurable null set $Z$ with $N\subseteq Z$.

Apply Tonelli in [L2] to ${\mathbf{1}}_Z$. Since $(\mu \times \nu )(Z)=0$, it gives ${\int }_X\nu (Z_x)d\mu =0,{\int }_Y\mu (Z^y)d\nu =0.$ The functions $x\mapsto \nu (Z_x)$ and $y\mapsto \mu (Z^y)$ are measurable, so $A:=\{x:\nu (Z_x)>0\}$ and $B:=\{y:\mu (Z^y)>0\}$ are measurable null sets. For $x\mathrm{\notin }A$, $f_x=g_x$ outside the null section $Z_x$; [L4] makes $f_x$ measurable for the completed factor measure. The analogous statement holds for $y\mathrm{\notin }B$.

In the nonnegative case, replace the product-measurable representative $g$ by $g^{+}=\max (g,0)$. Since $f\ge 0$ and $f=g$ outside $Z$, one has $f=g^{+}$ there, both on the product and on every section outside $A,B$. Apply Tonelli [L2] to $g^{+}$. Define $I_f$ and $J_f$ to equal the section integrals of $f$ outside $A,B$ and zero on $A,B$ as in the statement. They agree there with the measurable Tonelli section-integral functions of $g^{+}$, so are measurable and have the same outer integrals. By [L3], the completed integral of $f$ equals the product integral of $g^{+}$. Tonelli now gives the asserted three-way equality.

If $f\in L^1(\overline{\mu \times \nu })$, apply [L1] separately to $\mathrm{Re}f$ and $\mathrm{Im}f$, obtaining product-measurable extended-real functions $u,v$ equal to those finite parts almost everywhere. Their nonfinite-value sets are product-null; set $u,v$ to zero there, making them finite real-valued product-measurable functions. Put $g:=u+iv$. The discrepancy set $\{f\mathrm{\ne }g\}$ is contained in a product-measurable null set $Z$ by the completion definition, so $g=f$ outside $Z$. By [L3] applied to $\mathrm{\mid }f\mathrm{\mid }$ and $\mathrm{\mid }g\mathrm{\mid }$, $g\in L^1(\mu \times \nu )$. By Fubini [L2], the measurable sets where $\int \mathrm{\mid }{g}_x\mathrm{\mid }d\nu =\mathrm{\infty }$ or $\int \mathrm{\mid }{g}^y\mathrm{\mid }d\mu =\mathrm{\infty }$ are factor-null. Apply the Tonelli argument of step 2.1 to this new $Z$, obtaining measurable factor-null sets $A,B$, and enlarge them by these nonintegrability sets. Outside the enlarged sets, apply [L4] separately to the real and imaginary parts to make each $f$ section measurable; its almost-everywhere equality to the corresponding integrable $g$ section then makes it integrable with the same integral. Define $I_f,J_f$ by these section integrals there and zero on the exceptional sets. They are measurable, integrable, and equal almost everywhere to the Fubini section-integral functions for $g$. By [L3], their integrals equal the completed product integral of $f$.