The following observation demonstrates the necessity of the integrability assumption in Fubini's theorem. Let $$ Q= \{ (x,y)\in \mathbb{R}^2: x\geq0, y\geq 0\ $$ denote the upper, right quadrant. Let $R\subset Q$ be the region in the quadrant bounded by the lines $y=x, y=x-1$ , and let let $S\subset Q$ be a similar region, but this time bounded by the lines $y=x-1,\; y=x-2$ . Let $$ f = \chi_{S}- \chi_R $$ where $\chi$ denotes a characteristic function.

Observe that the Lebesgue measure of $R$ and of $S$ is infinite. Hence, $f$ is not a Lebesgue-integrable function. However for every $x\geq 0$ the function $$ g(x) = \int_0^\infty f(x,y)\, dy $$ is integrable. Indeed,