Talk:Radon–Nikodym theorem

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So presumably not just a function in the statement, but integrable? Charles Matthews 11:16, 11 Oct 2003 (UTC)

Yes, I'll make this explicit in the article. Pete 12:39, 11 Oct 2003 (UTC)

Another question is: f is not unique (actually, it is unique almost sure), as far as I know (but I may be perfectly wrong). If this is the case I think it is worth mentioning it (something like the equality holds for f and g whenever f=g a.s.). Pfortuny 12:48, 11 Mar 2004 (UTC)

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The second LaTeX formula bothers me. First of all, what is A supposed to be? I would guess a random variable rather than a set (like above). Also, shouldn't it rather be

(E_P and E_Q switched?). Not sure though, else I would change it myself. DrZ 13:50, 16 Mar 2004 (UTC)

I changed the last bit, presumably A is supposed to be same arbitrary measurable set as from the first formula? Pete/Pcb21 (talk) 14:10, 16 Mar 2004 (UTC)

But what is the expectation operator applied to a set supposed to mean? DrZ 14:50, 16 Mar 2004 (UTC)

That was wrong. I think now it is right. There is no expectation operator applied to a set, so you were right in your concern. Pfortuny 15:15, 16 Mar 2004 (UTC)

[edit] proof

heh, article claims theorem is from functinal analysis yet doesn't give a (more elegant) functional analytic proof. Mct mht 11:49, 10 June 2006 (UTC)

[edit] measures on an algebra (not sigma)

definition: ν <<< μ : for all e>0 exists d>0 for all A in algebra (μ(A)<d -> ν(A)<e)

lemma: (ν <<< μ) -> (ν << μ) lemma: for sigma additive ν, μ on a sigma-algebra (ν << μ) -> (ν <<< μ)

definition: simple-function : finite linear combination of indicator functions of measurable sets.

there's also a theorem that states: let ν <<< μ positive meaures on an algebra, let e>0 then exists a simple-function f s.t. ||ν-fdμ||<e.

with completeness of L1(μ), the regular Radon-Nikodym theorem follows. --itaj 00:22, 15 June 2006 (UTC)

I don't understand what you are trying to say. But either way I don't think this belongs in this article, rather a new article containing this should be created. Oleg Alexandrov (talk) 03:07, 15 June 2006 (UTC)

it looks like itaj is saying the following: <<< is a continuity condition that's equivalent to the usual absolte continuity <<, for countably additive measures. the notation ||.|| looks like it means the total mass of a measure. a suitable f is then obtained by approximation using simple functions. question is: when the assumption that L¹ being complete is invoked, aren't you implicitly extending to a σ-algebra? Mct mht 03:25, 15 June 2006 (UTC)

what i meant was to discuss that theorem i described. to see if it's to be added here or maybe make another article. and the last comment says that this theorem together with the theorem that L¹ is complete can prove the regular radon-nikodym theorem. --itaj 20:59, 15 June 2006 (UTC)

well, that's the question. when you use the assumption that L¹ is complete, implicitly you're extending to a sigma algebra anyway, no? in other words, the support of functions in L¹ might as well be addded to the algebra to make it a sigma algebra. so maybe it's not so much of an improvement. Mct mht 21:06, 15 June 2006 (UTC)

[edit] Proof for signed measure

The article says, "If ν is a signed measure, then it can be Hahn–Jordan decomposed as ν = ν+−ν− where one of the measures is finite". Yes, this is true. However, how can we be sure that the non-fininite measure is σ-finite? Is it always true? If yes, then any nonnegative measure is σ-finite? Jackzhp 02:34, 10 December 2006 (UTC)