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In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference.

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KL-divergence and sub-$\sigma$-algebras

I am trying to understand if the following claim is true: Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\emptyse …