I'm very new to undergraduate Lebesgue measure and am still having problems with it.
I'm trying to prove the following:
Let $A\subset\mathbb{R}^m$ and $B\subset\mathbb{R}^n$ and $A\times B\subset\mathbb{R}^{m+n}$.
If $A$ and $B$ are both measurable, then $A \times B$ is measurable.
If $A$ and $B$ are both measurable, $|A \times B|=|A|\cdot|B|$.