I'd like to show that, given borel measurable sets $\displaystyle A,B$ in $\displaystyle \mathbb{R}^{d_1}$ and $\displaystyle \mathbb{R}^{d_2}$ that

$\displaystyle A\times B =
\left\{(x,y): x\in A, y\in B\right\}$

is Borel-measurable set in $\displaystyle \mathbb{R}^{d_1+d_2}$

Although it seems intuitively clear, I find it hard to prove. The structure of the Borel sets A,B may be quite complex, so I don't see how I could start. Can anyone maybe offer some insight?