Application of Fubini's theorem

Question

Find a Lebesgue integrable function $f:[-1,1] \times [-1,1]\rightarrow \mathbb{R}$ such that $$ \int_{-1}^a\int_{-1}^1 f(x,y) \ dy \ dx=0= \int_{-1}^1\int_{-1}^b f(x,y) \ dy \ dx $$ for every $a,b \in [-1,1]$ and such that $\{(x,y): f(x,y)=0\}$ has measure zero.

I started with $f(x,y)=1_{\mathbb{Q}\cap[-1,1]}(x).1_{\mathbb{Q}\cap[-1,1]}(y) $ but this violates the last condition that measure of set where $f$ attains zero is $0$. Can anyone help me with this ?

Answer 1

Take the function $f(x,y)$ to be of the form $g(x)h(y)$ for two Lebesgue integrable and odd function $g$ and $h$. We have to grantee zero-measure set of function's zeros. We can take a PDF of a Rayliegh distributed random variable extended over $\mathbb R$ and then $f(x,y)=xye^{\frac{-x^2-y^2}2}$ which yields the desired function.

Answer 2

$f$ defined on $[-1,1]\times[-1,1]$ by:

$$f(x,y)=\begin{cases} -1 \text{ for } xy \ge 0\\ 1 \text{ else} \end{cases}$$

works.