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What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...

If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in ...

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...