Convex Stochastic Duality and the "Biting Lemma"

A standard approach to duality in stochastic optimization problems with constraints in $L_{\infty}$ relies upon the Yosida - Hewitt theorem. We develop an alternative technique which employs only "elementary" means. The technique is based on an $\varepsilon$-regularization of the original problem and on passing to the limit as $\varepsilon \to 0$ with the help of a simple measure-theoretic fact -- the biting lemma.