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May 27

Resumed Discussion of Reprised Confusion over Wikipedia Definitions of Sigma Additivity

I have added the following reply to the thread discussing my reprised confusion over Wikipedia definitions of sigma additivity:

Hey, guys; I'm sorry that I haven't replied to this thread even though I know that I should have done so due to the fact that I'm the person who started it, but I've sort of been mulling over exactly how to continue this discussion. I've been having a little trouble figuring out exactly what you said, Straightontillmorning, even though it was easy to understand at first glance. I think that the problem is that I couldn't exactly bring myself to formulate the questions that I wanted to ask you (they were on the tip of my tongue&thinsp–or rather, my fingers, I guess….) Regardless, however, of the trouble that I temporarily had in communicating with you, I think that I'm ready to discuss your comments. First of all, I comprehend that you're trying to tell me that I phrased my statement that 'the definition of sigma additivity can apply to either a probability measure $P$ or a generic measure $μ$ ' incorrectly and should instead have made clear that I then understood that the definition of sigma additivity applies to both probability measures $P$ and general measures $μ$ because the former are defined in terms of the latter, albeit with the extra condition that a probability measure must output a result between 0 and 1. As such, the definitions are, as you said, almost identical. Second of all, are the 'hypotheses' that you list as bullet points below your first paragraphs simply summaries of the conditions applied respectively to each of the definitions that I took from the articles that I referenced? If so, then could you please explain to me what exactly the difference is between a sequence, collection, a set of pairwise disjoint sets, a multiset such as the kind that one must derive from such groups of elements if some of these elements are identical but must not be allowed to collapse into a single, merged element, or an indexed family of sets. Thirdly, I understand that when you mention a 'countable sequence' that you mean that one would use a finite index set to index the sequence in question. Fifth, I get that one may use either any capital letter $A,B,\dots ,Z$ or any subscript attached to the letter 'E' – as in $E_{1},E_{2},E_{3},\dots ,E_{n}$ – to denote an event and either the Greek letter Sigma $Σ$ or a calligraphic letter such as ${\mathcal {A}},{\mathcal {B}},{\mathcal {C}},$ et cetera, to denote a sigma algebra. And, sixth and finally, I find it preposterous that one could equate the arbitrary, but still finite, union used in the first two definitions with the infinite union used in the last definition. I hope that that helps you understand why I'm still a little confused.

Sincerely,

RandomDSdevel (talk) 19:28, 27 May 2013 (UTC)
— RandomDSdevel (talk) 19:34, 27 May 2013 (UTC), Archived post concerning my Reprised Confusion over Wikipedia Definitions of Sigma Additivity

Please respond to it over at the respective archives page.
Thanks in advance,
RandomDSdevel (talk) 19:34, 27 May 2013 (UTC)[reply]

TLDR. May I suggest that your confusion might be more easily resolved by a visit to your local library instead of repeatedly trying to get answers from a reference desk. A standard reference for measure theory is Halmos "Measure theory". A more readable account is Royden "Real analysis". Sławomir Biały (talk) 21:44, 27 May 2013 (UTC)[reply]