Mathematics – Functional Analysis
Scientific paper
2011-08-22
Mathematics
Functional Analysis
Scientific paper
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.
Avigad Jeremy
Dean Edward
Rute Jason
No associations
LandOfFree
Metastable convergence theorems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Metastable convergence theorems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metastable convergence theorems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-176803