User:RandomDSdevel/sandbox/Upcoming Wikipedia Mathematics Reference Desk Archive Reply

Continuation of Thread on 'Reprised Confusion over Wikipedia Definitions of Sigma Additivity'

Sorry for not being able to respond for a while again, Straightontillmorning and, I guess, whoever else may be paying attention to this thread, but I have been having a little more trouble getting on to my family's computer since everybody else's school year ended (I'm taking an extended one, but methinks that this might be somewhat beside the point by now…) On top of that, I had to leave on vacation for a week in the middle of last month, and this obviously didn't help my ability to answer your reply. Anyhow, I suspect that you're getting a little tired of me having to explain why I haven't been able to contact you every time I post, so I'll try to return to the punctuality that I originally had in responding to this thread from now on. As for actually replying, however, I should probably get started, so here goes: first, does the 'Ok' (sic) that began your last post mean that you understand how I interpreted the relationship between generic measures and more strictly defined probability measures? If so, then I'm glad that we've finally gotten that misunderstanding cleared up. Second, thank you for clarifying that the 'hypotheses' that, as I had guessed in the following quote from the question that originally prompted this clarification, you had listed '…as bullet points below your first paragraphs…' were '…simply summaries of the conditions applied respectively to each of the definitions that I took from the articles that I referenced….' Third, however, I must admit that I found your explanation of the differences between sets, collections, multisets, non-indexed families of sets, indexed families of sets, and sequences to be somewhat confusing. As such, I went to double-check your facts against those posted in the articles that Wikipedia and Wolfram MathWorld only to find that my suspicions that you got everything mixed up were correct! To begin with, a collection is, according to its description, equivalent to a multiset, not a set. As such, a multiset, a collection, and, thereby, a family of sets (see the first sentence in the linked article,) indexed or not, are all just different names for the same concept. You were, however, correct, in stating that a sequence is simply an indexed family of sets indexed by the set $\mathbb {N}$ of all natural numbers. As such, I can safely conclude that the facts that a collection is actually a multiset and that I must put all of the data with which my textbook requires me to work into multisets because of how I would otherwise lose data justifies my introduction of the concept of a multiset. Fourth, I understand that a countable set is a set whose cardinality is less than or equal to that of the set $\mathbb {N}$ of natural numbers such that…