Suppose $\mathcal{F} = \{f_x : x \in \mathbb{R}^d \}$ and each $f_x$ shares the same law $P$. If $\mathcal{F}$ is a class of uniformly bounded functions satisfying $L_r$-continuity, i.e. $\forall f \in \mathcal{F}$, we have

$\| f\|_{\infty} : = \max_{x \in \mathbb{R}^d} |f(x)| < \infty$.

$\mathrm{E} \sup_{|x - y| \leq \delta} |f_x - f_y|^r \leq \delta^{v}$,

then the article said the covering number of $\mathcal{F}$ satisfies $$N \big(\epsilon, \mathcal{F}, L_r(P) \big) \leq C \left( \frac{1}{\epsilon} \right)^{\frac{d}{v}}$$ for some positive constant $C >0$.

I am extremely confused about this result.

Could anyone help me? Thanks in advance.