A metastable dominated convergence theorem
Jeremy D. Avigad, Dean, Rute · Journal of Logic and Analysis · 2012
The dominated convergence theorem implies that if (f n ) is a sequence of functions on a probability space taking values in the interval [0, 1], and (f n ) converges pointwise a.e., then ( f n ) converges to the integral of the pointwise limit.Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (f n ) and the underlying space.We prove a slight strengthening of Tao's theorem which, moreover, provides an explicit description of the second bound in terms of the first.Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences.We also establish a quantitative version of Egorov's theorem, and introduce a new mode of convergence related to these notions.