First of all, I am sorry if this seems basic. I just don't know where to begin.
Let $X=[0,1]$. Let $\mu$ be the Lebesgue measure. Consider the functions $g_1=1_{[0,1/2]}~,g_2=1_{[1/2,3/4]}~,g_3=1_{[3/4,7/8]}~\ldots$ The question I want to ask is whether or not the function $f(x,y) = \sum_{n=1}^\infty [g_n(x)-g_{n+1}(x)]g_n(y)$ is integrable on $[0,1]\times [0,1].$