A theorem on majorizing measures

Let $(T,d)$ be a metric space and $\varphi :{\mathbb{R}}_{+}\to \mathbb{R}$ an increasing, convex function with $\varphi (0)=0$. We prove that if $m$ is a probability measure $m$ on $T$ which is majorizing with respect to $d,\varphi$, that is, $$\mathcal{S}$:=\sup_{x\in T}\int^{D(T)}_0\phi^{-1}(\frac{1}{m(B(x,\epsilon))}) d\epsilon <\infty$, then $$\mathbf{E}\underset{s,t\in T}{\sup }\mathrm{\mid }X(s)-X(t)\mathrm{\mid }\le 32\mathcal{S}$$ for each separable stochastic process $X(t)$, $t\in T$, which satisfies $\mathbf{E}\varphi (\frac{\mathrm{\mid }X(s)-X(t)\mathrm{\mid }}{d(s,t)})\le 1$ 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.