A theorem on majorizing measures

Abstract

Let (T,d) be a metric space and φ:ℝ₊→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, $\mathscr{S}:=\sup_{x\in T}\int^{D(T)}_{0}\varphi^{-1}(\frac {1}{m(B(x,\varepsilon ))})\,d\varepsilon <\infty$, then $$\mathbf{E} \sup_{s,t \in T}|X(s)−X(t)|\le32\mathscr{S}$$ for each separable stochastic process X(t), t∈T, which satisfies $\mathbf{E}\varphi(\frac {|X(s)-X(t)|}{d(s,t)})\leq 1$ for all s, t∈T, s≠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.