Suppose $\{I_n\}$ is a pairwise disjoint sequence of sub intervals of $[0,1]$ of positive length. For each $n$ let $a_n$ be the reciprocal of the length of $I_n$, and let $g_n$ be the characteristic function of $I_n$ multiplied by $a_n$. Define $f$ by $$f(x,y)=\sum_{n=1}^{\infty} [g_n(x)-g_{n+1}(x)]g_n(y)$$ for $0\leq x, y\leq 1$.
I have shown that $$\int_0^1 \int_0^1 f(x,y) dx dy = 0 \qquad \int_0^1 \int_0^1 f(x,y) dy dx = 1 $$
I now need to explain why this does not contradict Fubini's Theorem. My guess is that $|f(x,y)|$ is not integrable, but I'm not sure how to calculate that integral to prove that.
$$\int_{[0,1]\times[0,1]}|f(x,y)|\operatorname{d}x\operatorname{d}y\ge\int_{\cup_{n\in\mathbb{N}}I_n\times I_n}|f(x,y)|\operatorname{d}x\operatorname{d}y\\ =\sum_{n\in\mathbb{N}}\int_{I_n\times I_n}|f(x,y)|\operatorname{d}x\operatorname{d}y = \sum_{n\in\mathbb{N}}\int_{I_n\times I_n}\frac{1}{|I_n|^2}\operatorname{d}x\operatorname{d}y = \sum_{n\in\mathbb{N}} 1 = +\infty$$