I am interested in the $\| \cdot \|_{\infty}$-norm bracketing number or covering number of some collection of distribution functions on $\mathbb{R}$.
Let $\mathcal{F}$ consist of all distribution functions on $\mathbb{R}$. There is an existing result (Thm 2.7.5, van der Vaart and Wellner, 1996) on the $L_r(Q)$-norm bracketing number: For any $1 \leq r<\infty$ and any probability measure $Q$, there exists a constant $K$ depending only on $r$ that $$ \log N_{[]}(\varepsilon, \mathcal{F}, L_r(Q) ) \leq K \frac{1}{\varepsilon}. $$ However, using $\|\cdot\|_{\infty}$-norm, the result does not hold. In fact, $N_{[]}(\varepsilon, \mathcal{F}, \|\cdot\|_{\infty}) = \infty$ for $\varepsilon<1/2$.
To have a nontrivial upper bound, if possible, I consider $$ \mathcal{F}_s = \{ F \in \mathcal{F} : f \text{ is $m$-times continuously differentible, } \|f\|_{C^m} \leq 1 \} $$ where $\|f\|_{C^m} := \sum_{j=0}^m \|f^{(j)} \|_{\infty} $.
My questions is, what do we know about the covering number or bracketing number, under $\|\cdot\|_{\infty}$ norm, of such $\mathcal{F}_s$? Are they finite? If finite, is there any nontrivial upper bound?
I am aware of the results in Chapter 2.7 of van der Vaart and Wellner (1996). However, none of them is suitable, since I need the functions to have unbounded domain, and to be covered or bracketed in the supremum norm. This question appears to be relevant, but it only assumes Lipschitz continuity, and I couldn't obtain finiteness using Lemma 6 in the paper mentioned therein.
