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Ah, yes, nonnegativity—I was afraid I was overlooking something. Very well, then I agree that there is a place for as many articles as currently exist and I withdraw my merge nomination. I'm still confused though: is this meant to be a page about σ-additivity (the axiom which may or may not be used
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to an independent article, but I don't understand why, since that article also has as its major topic a definition and elementary properties of countable additivity. It's possible that the content here is easier to follow for the uninitiated, but having two definitions in different articles is just
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Ok added example, but its very late here, and it probaby contains some kind of error. It definitly needs to be written more clearly.. still, it works (I think). It basically tries to fix the original example given in such a way as to not allow for sets which aren't intervals around 0, and thus you
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once this article was written. That link is critical: it will encourage editors to expand/expound in this article, not the other. Twice, now, in two days, I've written expanded sections in one article only to discover shortly afterwards that what I wrote was already covered in another article,
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When I follow a link of some term like "sigma additivity", because I want to know what it means, I would rather be sent to an article just about that term, which defines that term right at the top, instead of being directed to an article about a different topic and having to read down into the
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Oops, my adjustment only works by only considering intervals as subsets of R. I has managed to make sure that you could only consider an interval in order to get a measure 1 set, however as it was defined on the powerset of R (i.e. all possible subsets of R), you can have a=(0,1)-rationals and
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in a context with other axioms) or about the class of σ-additive functions? Especially from your argument 4 (and the page title) I infer the former, but (assuming that σ-additive functions which aren't measures are ever notable) shouldn't we have a separate page for the latter? And
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It might be useful to explain that the measure is often taken to be non-negative, but that the rpinciples of sigma-additivity work just as well for an extended (negative) measure as well. Just so that those of us expecting a positive measure aren't surprised.
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I would be fully convinced into agreeing with you both if I saw evidence that the notion of σ-additivity has application outside the definition of a measure specifically (or at least, outside trivial generalizations thereof). My concern is precisely that it
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The (current) example of an additive function on the powerset of the reals that is not sigma additive is wrong. By the given formula, the measure of the irrationals is 1 and the measure of the rationals is 1, but the measure of their union is again 1.
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To fix this problem I had considered trying to define the function on only intervals of R, I can't actually remember if that is allowed in terms of /mu being a measure.. but then Is suppose this is only a general function... hmm could someone check that
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Fibonacci, you know very well that a measure can't take both +∞ and −∞ as values, as then you can't add up the two. You either explain this in the article, or you restrict yourself to an interval not containing both of those.
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842:-- Yeah, I agree this example is so blatently flawed it should be removed.. I will put your edit in for now, and attempt to come up with a different example next time Im bored... --
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But the point of the question was that the article was written as if a σ-additive function could only take values in (−∞, +∞]; I changed it to the whole extended real line, and
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but that's OK. I don't really like super articles containing a lot of stuff when you just want to look up a single thing. That is, I'd vote to keep this article separate.
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And you know that the definition allows it (in principle) to take both values, but then it is proven that you cannot possibly have both. I'd go for the former option. --
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have a section on "Generalizations": your philosophy expressed in 4 seems to suggest it doesn't belong; I feel that it's fine (and should include a sentence linking to
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here. However I really am by no means an expert in this area and I would appreciate, the views of more knowledgeable editors. I will see if I can round some up ;-)
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But that makes the function non-additive, see the section "additive but not sigma-additive" above. I've now fixed this (I hope).
810:{\displaystyle \mu (A)={\begin{cases}\infty &{\mbox{ if }}0\in {\bar {A}}\\0&{\mbox{ if }}0\notin {\bar {A}}\end{cases}}}
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I created this article because I thought that the topic "sigma additivity" was deserving of its own article. Here are my reasons:
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you'll see that σ-additivity is the only axiom, and hence "σ-additive function" and "measure" are actually the same concept. —
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Yeah, one could, but this defintion is not so interesting. You could as well deal with A being a sigma-algebra to start with.
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This article has the possibility of expanding further beyond what would necessarily be appropriate for the article
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So... again, why did you revert my edit? Why do you think it gives "nothing but headaches"? Please answer here. --
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May I ask the purpose of the degree symbol appearing after the parenthetic describing your clarification?
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page. Anyway, those are my thoughts, I don't want to contest this more, do whatever you think best. —
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So for these reasons I would prefer to leave this article here. I have, for the time being, redirected
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There ought to at least be at least an explicit mention of Finite
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Agree with Oleg & Paul. However, one mistake was to not link sigma-additive in the the article
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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There ought to at least be at least an explicit mention of Finite
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in my opinion, the word "infinity" in this sentence must be replaced with the word "one".
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The following is not additive, not when defined on the power set of the reals anyway.
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actually defines a "sigma additive function", rather it defines a "countably additive
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restricting the function to the power set of the positive reals won't work either...
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The definition of the property in this article is more general than the one given in
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Hmmm... I see you've intentionally and knowingly changed this from a redirect to
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asking for them to be expanded in independent, overlapping and confusing ways.
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Kolmogorov's original defining axiom AIUI was that for any sequence
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I agree with Paul's reasoning. There is of course some overlap with
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Not only (nonnegative) measures are σ-additive. Measures have
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In short, I move that this content be merged back into
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73:Low‑priority
51:WikiProjects
1570:Paul August
1391:Paul August
1141:—Preceding
1120:—Preceding
418:Paul August
354:Paul August
293:Paul August
123:Mathematics
114:mathematics
70:Mathematics
41:Start-class
1587:Categories
1365:ciphergoth
1205:such that
1086:-unsigned
874:please...
516:Fibonacci
472:Fibonacci
434:Fibonacci
1305:we have
1143:unsigned
1122:unsigned
919:X a set
894:unsigned
381:Blotwell
367:already
337:Blotwell
226:Blotwell
180:unsigned
333:measure
258:measure
150:on the
47:scale.
1251:and
986:then
879:TM-77
859:TM-77
844:TM-77
445:linas
408:linas
1405:from
1369:talk
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1130:talk
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545:talk
494:talk
369:does
188:talk
1523:if
1443:if
1314:lim
932:if
778:if
744:if
641:if
607:if
260:".)
142:Low
1589::
1540:¯
1531:∈
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1438:∞
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1329:μ
1324:∞
1321:→
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1275:∞
1260:⋃
1223:⊃
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877:--
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1573:☎
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1267:=
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1237:1
1234:+
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1227:A
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1190:A
1177:2
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1065:A
1061:(
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1008:=
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974:A
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951:1
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900:(
792:A
783:0
771:0
758:A
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731:{
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717:(
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580:(
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357:☎
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186:(
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53::
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