A theorem on majorizing measures

Abstract

Let $(T,d)$ be a metric space and $ϕ:\mathbb{R}_+\to \mathbb{R}$ an increasing, convex function with $ϕ(0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,ϕ$, that is, $\mathcal{S}:=\sup_{x\in T}\int^{D(T)}_0ϕ^{-1}(\frac{1}{m(B(x,ε))}) dε<\infty$, then \[\mathbf{E}\sup_{s,t\in T}|X(s)-X(t)|\leq 32\mathcal{S}\] for each separable stochastic process $X(t)$, $t\in T$, which satisfies $\mathbf{E}ϕ(\frac{|X(s)-X(t)|}{d(s,t)})\leq 1$ for all $s,t\in T$, $s\neq t$. This is a strengthening of one of the main results from Talagrand [Ann. Probab. 18 (1990) 1--49], and its proof is significantly simpler.