Talk:Measurable function

message: Borel function - that para really needs turning round to be clear.
Charles Matthews 12:48, 6 Sep 2004 (UTC)
thread_title: Borel function

message: From the article:
A useful characterisation of Lebesgue measurable functions is that f is measurable if and only if mid{-g,f,g} is integrable for all non-negative Lebesgue integrable functions g.
Why is it useful?  What does mid{-g,f,g} mean? Timhoooey 04:02, 14 October 2007 (UTC)
thread_title: What is mid{-g,f,g}?

message: The article says that f : X → Y is measurable provided that for every subset Z of Y in T the preimage of Z is in Σ.  Two questions... first, nobody said f was surjective, so in general, Z doesn't even have a preimage.  Second, nobody said f was injective, so how can we talk about the preimage of Z?  Am I missing something obvious?—PaulTanenbaum (talk) 01:26, 28 February 2008 (UTC)
replies: text: The preimage of a set Q under a function f is the set of points x in the domain such that f(x) is in Q. This is well defined (and, in particular, unique) for every subset of the codomain of the function. It may be that the preimage of Q is the empty set, for example, if no point in the domain maps into Q. — Carl (CBM · talk) 01:56, 28 February 2008 (UTC)
replies: text: Ah, I was unacquainted with the notion of preimage of a set so, in case it's not clear from my original question, I presumed that f-1({y1, y2, ...}) was being used to mean {f-1(y1), f-1(y2), ...}.—PaulTanenbaum (talk) 04:23, 28 February 2008 (UTC)
replies: text: For a countable set, that is exactly the right idea, with the caveats that if f-1(yk) doesn't exist for some k (because yk isn't in the range) then there is no contribution to the preimage set, and if multiple points map to yk then they all get put into the preimage set.  — Carl (CBM · talk) 14:34, 28 February 2008 (UTC)
thread_title:  Glitch in definition? 

message: because it duplicates some of the material in measure-preserving dynamical system. (Moreover, there was an inconsistency,
since the assumption that the measure is a probability measure was implicit, but not explicit.)  —Preceding unsigned comment added by OdedSchramm (talk • contribs) 16:42, 8 April 2008 (UTC)
thread_title: Deleted Section "Stationary transformations"

message: I usually come to wikipediate maths pages looking for a specific result or definition. Therefore I think it best to include a section that contains a formal definition. Removing the definition from the introductory paragraph would also allow this paragraph to provide a more intuitive idea of the concept. BenWhale (talk) 22:44, 12 July 2010 (UTC)
thread_title: Separate section with formal definition?

message: As one poster said earlier, this pages suffers from excessive generality.  I am studying analysis in graduate school and this page is so convoluted it had me questioning what it meant to be measurable.  Could we rewrite the page to focus on Lebesgue measurability and one or more of the definitions for that [e.g. f^{-1}( (x,\infty)) is measurable for all a or f^{-1}(B) is measurable for all Borel sets B], then move on to definitions in full generality? Ckhenderson (talk) 21:58, 20 July 2010 (UTC)
thread_title: Rewrite?

message: A function f:X->R is said to be measurable if, for every \alpha \in R,  {x \in X : f(x) > \alpah} \in Sigma.
Sigma is the sigma-algebra which contains X.  from Linear Functional Analysis by B.P. Rynne.  Brydustin (talk) 00:20, 8 February 2012 (UTC)
thread_title:  Different Definition? 

message: There is a notion of a function being -measurable. At least I have seen it for outer measures. Let  be a space with outer measure  and let  be a topological space. A function  is -measurable if the inverse image of every open set is -measurable. (Perhaps  is required to be Borel regular, don't remember.) No it isn't. Then it is a theorem that  is -measurable if and only if the inverse image of every Borel set is -measurable.
Now, as far as I can see, if  and if , then the above reduces to the content of the section Caveat.
The notion of -measurable function makes much sense, at least if  is a topological space and  is Borel regular. I'm not too sure whether the distinction between measure and outer measure is devastating. At any rate, there is an outer measure bearing the name Lebesgue measure that reduces to (ordinary) Lebesgue measure when restricted to the -algebra of measurable sets.
Provided my observations make sense, perhaps the article could introduce -measurable and thereby put Lebesgue measurable function in a context. A possible reference is 
* 
YohanN7 (talk) 10:01, 11 July 2016 (UTC)
thread_title:  Measurable and μ-measurable functions 

message: I'm going to re-write the lede because the current one, in my view, doesn't serve its purpose. First, the opening sentence (i.e. "a measurable
function is a structure-preserving function between measurable spaces") is misleading. Second, there is some extraneous fluff (e.g. "... functions that are not Lebesgue measurable are generally considered pathological"). Third, and most importantly, the lede in this particular article probably shouldn't be written in Scientific American style and try to dumb the subject down.  In other articles ("Probability Theory" is a good example of that), having a SciA-like intro is great. But measurable function just seems too specialized a notion to be of interest to the non-geeky crowd. StrokeOfMidnight (talk) 21:53, 3 August 2017 (UTC)
thread_title:  Re-writing the lede 

message: A discussion is taking place to address the redirect :Bimeasurable function. The discussion will occur at Wikipedia:Redirects for discussion/Log/2021 January 11#Bimeasurable function until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 13:09, 11 January 2021 (UTC)
thread_title:  "Bimeasurable function" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]