A clarification about Fubini's theorem

Question

Suppose $f(x,y)$ is a measurable function defined on an open set $G\subset\mathbb{R}^{m}\times \mathbb{R}^{n}$ ($m,n\geq1$), and $E\subset\mathbb{R}^{m}$ and $F\subset\mathbb{R}^{n}$ are two compact sets such that $E\times F\subset G$. Suppose
$$g(y)=\int_{E}f(x,y)dx\geq0$$ and is integrable on $F$, i.e., $$\int_{F}|g(y)|dy=\int_{F}g(y)dy<\infty.$$ Can we say that $f$ is integrable on $E\times F$, i.e., $$\int_{F}\int_{E}|f(x,y)|dxdy<\infty?$$

Answer 1

No, you cannot draw such a conclusion. Just take $f(x,y)=x h(y)$ where $h$ is not integrable. If $E$ is symmetric then $g(y)=0$ for all $y$ but $f$ is not integrable.