# A detail on Fubini's theorem

Let $f(x, y)$ be a measurable function on a product of two balls $B_{1}$ and $B_{2}$ in $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$ respectively and $m,n\geq1$. We know, according to Fubini's theorem, that if $f$ is non-negative, then the iterated integrals $$\int_{B_{1}}\int_{B_{2}}f(x,y)dxdy$$ and $$\int_{B_{2}}\int_{B_{1}}f(x,y)dydx$$ are equal in the sense that they both may be $+\infty$. My question is: Suppose that $f$ is not necessarily non-negative. If the iterated integrals are both finite, can they be different? Thanks.

• Wikipedia's article on Fubini's theorem is a good one, there is an example ($m=n=1$) which you need, that is, it shows that even if both iterated integrals are finite then they might be different en.wikipedia.org/wiki/… and here are the conditions which you need for the integrals to be the same en.wikipedia.org/wiki/… – m_gnacik May 22 '16 at 22:34