Measurability of the diagonal in the product space

Question

Let $(E,\mathcal E)$ be a measurable space. Under which assumption on $(E,\mathcal E)$ can we show that $\Delta:=\left\{(x,x):x\in E\right\}\in\mathcal E\otimes\mathcal E$? Note that this doesn't hold in general. Is it correct, for example, if $E$ is a Polish space and $\mathcal E=\mathcal B(E)$? A reference with a proof would be enough for me.

Answer 1

True for the Borel sigma algebra of any second countable space (in particular separable metric space).