Let $A$ be a rectangle in $\mathbb{R^k}$; let $B$ be a rectangle in $\mathbb{R^n}$; let $Q=A\times B$. Let $f: Q\to \mathbb{R}$ be a bounded function. Show that if $\int_Q f$ exists, then
$$\int_{y\in B}f(x,y)$$ exists for $x\in A-D$, where $D$ is a set of measure zero in $\mathbb{R^k}$.
My work
Let $Q’=((A-D)\times B)=(A\times B)-(D\times B)$. Then, I only need to show $D\times B$ has measure $0$ in $Q$.
Since it has measure $0$ is it integrable on $Q’$, thus by Fubini theorem it is $\int_{y\in B}f(x,y)$ exists $\forall x\in A-D$
Is this a correct way to thinking? And how can I say $D\times B$ has measure $0$ in Q?