On the expected values of sequences of functions

We prove new extensions to lemmas about combinations of convergent sequences of distribution functions and absolutely continuous bounded functions. New lemma one, a generalized Helly theorem, allows computing the limit of the expected value of a sequence of functions with respect to a sequence of measures. Previously published results allow either the function or the measure to be a sequence, but not both. Lemma two allows computing the expected value of an absolutely continuous monotone function by integrating the probabilities of the inverse function values. Previous results were restricted to the identity function. Lemma three gives a computationally and analytically convenient form for the limit of the expected value of a sequence of functions of a sequence of random variables. This is a new result that follows directly from the first two lemmas. Although the lemmas resemble standard results and seem obviously true, we have found only similar looking and related but quite distinct results in the literature. We provide examples which highlight the value of the new results.