In this paper we are interested in finding upper functions for a collection real-valued random variables Ψ χ θ , θ ∈ Θ. Here {χ θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a non-random function U : Θ → R+ such that sup θ∈Θ Ψ χ θ −U (θ) + is " small " with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to gaussian random functions and empirical processes.