Let $A,B \subset \mathbb{R}$ two bounded and Lebesgue measurable sets. I have to show that $A\times B \subset \mathbb{R}^2$ is measurable and \begin{align*}
\lambda(A \times B) = \lambda (A) \cdot \lambda (B)
\end{align*}
where $\lambda$ is the Lebesgue measure.
I tried to show the equality of the measure using the Fubini's Theorem with the indicators function but I'm not sure is the right way and I have no idea how to show that $A \times B$ is measurable.
Any suggestions? Thanks in advance!!
Best Answer
$A$ bounded is measurable if and only if for every $\epsilon >0$ there exists $U$ a finite union of intervals such that $m^{\star}(A\Delta U)<\epsilon $. Do this for $B$ too and get $$m^{\star}((A\times B) \Delta (U\times V))< 2M\epsilon $$ Multiplicativity should be easy now, since it works for $U$ and $V$.