Return to Revisions

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, $$|F_{Y|W}(y_2|W) - F_{Y|W}(y_1|W)| \leq \int 1(y_1 \leq s \leq y_2) P(Y = ds|W).$$ In particular, your function space is obtained by (I think) a Lipschitz transformation applied to the function space $$\mathcal{F}_{ind} = \{s \mapsto 1(s\leq y): y \in \mathbb{R}\}.$$ So your covering number is essentially that of $\mathcal{F}_{ind} $.