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> Is the diagonal $\Delta\_\mathbb{R}:=\{(x,x):x\in\mathbb{R}\}$ in the product $\sigma$-algebra $\mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})$, where $\mathcal{P}(\mathbb{R})$ is the power set of $\mathbb{R}$?
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I know that $\Delta\_{X}\notin\mathcal{P}(X)\otimes\mathcal{P}(X)$ if $|X|=2^{|\mathbb{R}|}$. (A proof can be found as an exercise in the first chapter of 
*Analysis III* by H. Amann and J. Escher.) It seems that whether $\mathcal{P}(\mathbb{R}\times\mathbb{R})=\mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})$ relies on set theoretic axioms like the continuum hypothesis. I wonder whether the diagonal is in $\mathcal{P}(\mathbb{R})\otimes\mathcal{P}(\mathbb{R})$?