Find an integrable $f$ on $[0, 1]^2$

Question

I can't find anyone can help me?

Find a real-valued function $f$ that is integrable on $[0,1]^2$ such that

$\int^{a}_{0} \int^{1}_{ 0} f(x, y) dy dx = 0, \\\int ^{1}_{0}\int^{b}_{0} f(x, y) dy dx = 0 $

for every $a,b \in [0, 1]$, but $f$ is not zero almost everywhere with respect to 2-dimensional Lebesgue measure.

Answer 1

Take a suitable function $g(x)$ not identically 0 st $\int_0^1g(x)\ dx=0$, and then take $f(x,y)=g(x)g(y)$. For example, take $$f(x,y)=\sin2\pi x\sin2\pi y$$

Clearly $f(x,y)=0$ only for $x=0,0.5$ or 1 or $y=0,0.5$ or 1. So the values of $(x,y)$ for which $f(x,y)=0$ are all on six line segments length 1 and hence have zero measure.