Product of Lebesgue measure of the cross is the multiplication of the measure

Question

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.

My first problem is to give a characterizion of $A\times B$ only knowing that $A,B$ are bounded.

I found some similitarity with this question Lebesgue Measure of the Cartesian Product