![]() Preliminaries and notationÄenote the power set of a set $X$ by $\wp(X).$ The upward closure or isotonization in $X$ of a family of subsets $\mathcal. The text below is copied from the Wikipedia article and somewhat edited. ![]() Is there a stronger definition of convergence that encapsulates both topological convergence, and almost sure convergence (and other similar notions)? I mean, if to call something a "convergence" in the "blablabla sense", my definition would have to satisfy some properties. ![]() I was then left wondering if there is a sort of "meta-definition" of convergence. The metaverse is controlled by large competing ecosystems - for example, Apple and Android meta worlds - with limited interoperability. Then, I learned, for my surprise, that almost sure convergence was not topological, hence, there are quite useful and natural ideas of convergence that are not covered by topological convergence. The metaverse remains a domain of niche applications, used by consumers for entertainment and gaming but stopping well short of an all-encompassing virtual reality. Similarly, if, for example, $X_n$ converged in probability to $X$, then this would be the same as $L^0$ convergence, which would imply a norm, which would imply a metric, which would imply a notion of convergence in the induced topology. If I had a metric space and defined convergence according to a metric $d$, then the topology induced by $d$ would match the topological notion of convergence. I thought that the topological definition of convergence was the most basic one in the following sense. ![]() So, just recently I realized that the idea of convergence is not "all encompassing". ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |