Władysław Wilczyński
Władysław Wilczyński: Convergence with respect to measure and category
17/03/21 09:33
Tuesday, March 23, 2021 17:00
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Władysław Wilczyński (University of Łódź)
Title: Convergence with respect to measure and category
Abstract: D. Fremlin in 1975 has proved that if \((X,S,m)\) is a probability space, then a sequence of measurable functions on \(X\) either has a subsequence convergent a.e., or there exists a subsequence without measurable pointwise cluster point. His proof is based upon the properties of weak convergent sequences in square integrable functions. The weaker form of the theorem was proved by Bucchioni and Goldman in1978. Their proof uses only some properties of the pair (family of measurable subsets of \([0,1]\), family of null sets). The pair (family of subsets of \([0,1]\) having the Baire property, family of sets of the first category) behaves similarly , so it was possible to obtain similar result for the convergence in category considered by E. Wagner in 1978.
Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.
Location: Zoom.us: if you want to participate please contact organizers
Speaker: Władysław Wilczyński (University of Łódź)
Title: Convergence with respect to measure and category
Abstract: D. Fremlin in 1975 has proved that if \((X,S,m)\) is a probability space, then a sequence of measurable functions on \(X\) either has a subsequence convergent a.e., or there exists a subsequence without measurable pointwise cluster point. His proof is based upon the properties of weak convergent sequences in square integrable functions. The weaker form of the theorem was proved by Bucchioni and Goldman in1978. Their proof uses only some properties of the pair (family of measurable subsets of \([0,1]\), family of null sets). The pair (family of subsets of \([0,1]\) having the Baire property, family of sets of the first category) behaves similarly , so it was possible to obtain similar result for the convergence in category considered by E. Wagner in 1978.
Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.