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

A\times B = <br />
\left\{(x,y): x\in A, y\in B\right\}

is Borel-measurable set in \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?