if $\int_Q f$ exists then $\int_{y \in B} f(x, y)$ exists for each $x \in A − D$, where $D$ is of measure zero. Let $f : Q = A \times B \to \Bbb R$ be bounded, where $A, B$ are respectively rectangles in $\Bbb R^l$ and $\Bbb R^k$. Show that if $\int_Q f$ exists then
$\int_{y \in B} f(x, y)$ exists for each $x \in A − D$, where $D$ is of measure zero.
Require Hints to solve the problem.

Hints: (if you are working with Riemann integration)
(1) For $f$ bounded and Riemann integrable on $Q = A \times B$ show that 
$$\int_Q f = \int_{x \in A} \underline{\int}_{y \in B}f(x,y) = \int_{x \in A}\overline{\int}_{y \in B}f(x,y)$$
where $\underline{\int}$ and $\overline{\int}$ denote lower and upper Darboux integrals.
(2) Since it must hold that
$$ \int_{x \in A} \left( \overline{\int}_{y \in B}f(x,y)-  \underline{\int}_{y \in B}f(x,y)  \right) = 0,$$
what can you conclude about where the upper and lower integrals are equal and consequently the existence of the integral $\int_{y \in B} f(x,y)$ for fixed $x$?
