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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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Covering number of the conditional distribution function

You need a different approach. Each function in your function space can be written as $$F_{Y|W}(y|W) = \int 1(s \leq y) P(Y = ds|W)$$ for some $y$. Thus, $$\|F_{Y|W}(y_2|W) - F_{Y|W}(y_1|W)\|_{L^1} = …
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