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Show that if $D=\{(x,x) \ | \ x \in \mathbb{R}\}, $ then $$D \in \mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})=\sigma(\{A\times B \ | \ A\subseteq \mathbb{R}, B\subseteq \mathbb{R} \}).$$ If we consider the space $(\mathbb R,\mathcal P (\mathbb R))$, then $\mathcal P(\mathbb R)=\mathcal B (\mathbb R).$

I know that for a second countable locally compact Hausdorff space $X$ $$\mathcal{B}(X)\otimes\mathcal{B}(X)= \mathcal{B}(X^2), $$ and from this the result I want would be obvious, but in this case we don't have that $(\mathbb R,\mathcal P (\mathbb R))$ is second countable. So I'm stuck. Any help would be greatly appreciated.

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    The Borel sigma algebra is contained in the power set of $\mathbf{R}$ and $\mathrm{D}$ es _closed_ in $\mathbf{R}^2$. Does this hint help? – 2017-01-04
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    No, sorry. Why does that help? – 2017-01-04
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    I have this idea that if $\mathsf{A}_1$ and $\mathsf{A_2}$ are sigma algebras contained in $\mathcal{A}_1$ and $\mathcal{A_2},$ respectively, then $\sigma(\mathsf{A}_1 \times \mathsf{A_2}) \subset \mathcal{A}_1 \otimes \mathsf{A}_2.$ If this were true, as $\mathcal{B}(\mathbf{R}^2) = \sigma(\mathcal{B}(\mathbf{R}) \times \mathcal{B}(\mathbf{R})),$ the exercise should follow. – 2017-01-04
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    I see it now. Thanks – 2017-01-04
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    @bof I edited the question, I hope it's a bit clearer now. – 2017-01-04

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$$\{(x,x)|x\in\mathbb R\}=\bigcap_{q\in\mathbb Q}\big[(-\infty,q)\times(-\infty,q)\cup[q,+\infty)\times[q,+\infty)\big]$$