Example of a bounded function $f : A\times B \to \mathbb{R}^n$ such that one of the iterated integrals exist but the other does not

I am studying Fubini theorem for riemann integrals and I was wondering if there are examples of a bounded function $f : A\times B \to \mathbb{R}^n$ such that $A \subset \mathbb{R}^n, B \subset \mathbb{R}^m$ are rectangles and such that one of the iterated integrals $\int_A f, \int_B$ f exist but the other does not, and even worst, if the both iterated integral exist but not the integral $\int_{A\times B} f$.

I do appreciate any suggestions, examples, etc!

Thanks a lot!

Hint: There exists $E\subset [0,1]^2,$ with $E$ dense in $[0,1]^2,$ such that each horizontal and vertical line intersects $E$ in at most finitely many points. Think about $f(x,y) = \chi_E(x,y).$
If we have $A=\mathbb{R}$, $B=\mathbb{R}$, take $f(x, y)=e^{-x^2}x$. We can show that $\int_{A}f(x, y)dx=0$. But $\int_Bf(x,y)dy=\lim_{n,m\to \infty}(n+m)f(x, y)\to \infty$.
For bounded $A$, $B$, we can also let $g(x)$ denote a nonintegrable function, then $g(x, y)=g(y)$ is constant in $X$, so $\int_{A}g(x, y)dx$ exists, but $\int_{B}g(x,y)dy=\int g(y)dy$ doesn't exist by assumption.