Metastable convergence theorems

TL;DR

This paper strengthens Tao's quantitative metastable convergence theorem for dominated convergence, providing explicit bounds and recursive computation methods, and extends these ideas to Egorov's theorem and a new convergence mode.

Contribution

It offers a strengthened, explicit version of Tao's metastable convergence theorem and introduces a new mode of convergence related to these quantitative bounds.

Findings

Explicit bounds for metastable convergence can be computed recursively.

A quantitative version of Egorov's theorem is established.

A new mode of convergence related to metastability is introduced.

Abstract

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 the sequence of integrals converges to the integral of the pointwise limit. Tao 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.