In a paper I am reading, the following result is used. If $(\mathcal{X},\mathcal{F}) $ is a measurable space, $\mathcal{G}$ is a class of functions with elements $f : \mathcal{X} \to \mathbb{R}$. We assume $\mathcal{G}$ is a bounded metric space. Then I wish to show that if

$$ \int_{0}^R\sqrt{H_B(u,\mathcal{G},L_2(P))}du < \infty $$

where $H_B(u,\mathcal{G},L_2(P))$ is the bracketing entropy, then

$$ \sup_{f \in \mathcal{G}} \Big| \frac{1}{n}\sum_{i=1}^nf(X_i)-\mathbb{E}(f(X)) \Big| \to 0 $$ in probability. Here, $R$ is a bound on the distance between functions in $\mathcal{G}$, i.e.

$$ \sup_{f_1,f_2 \in \mathcal{G}} d(f_1,f_2) < R. $$

I have looked various resources on empirical process theory, for example page 50 of http://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf#page50, but to no avail. The paper where it has been used is page 3 of https://projecteuclid.org/journals/supplementalcontent/10.1214/14-AOS1260/suppdf_1.pdf



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