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In topology, given topological spaces $X$ and $Y$, the compact-open topology is considered, under the relatively mild requirement that $X$ be locally compact Hausdorff, to be the most "natural" topology on the set $\mathcal{C}(X,Y)$ of continuous functions $X\to Y$. (I'm not going to even attempt to define "natural" here---take it to mean whatever seems most appropriate).

Is there something analogous, in the sense of being somehow "natural", for the set of measurable functions between two measure spaces?

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It depends on your perspective. The example you cite is taken primarily to study sequences (or nets) of functions with some pproperty P and hope that P in the closure or at least the limit. Are you also wanting to preserve such properties under limits, or are you looking to do algebraic or representational stuff with measurable functions? Gerhard "Ask Me About System Design" Paseman, 2012.08.08 –  Gerhard Paseman Aug 8 '12 at 21:18
    
Gerhard, I'm not really planning on doing anything specific with measurable functions. I'm just curious on what's out there. So, info about any of the perspectives you listed would be awesome. –  Avi Steiner Aug 9 '12 at 1:16
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2 Answers

up vote 9 down vote accepted

There is an impossibility theorem: If you let $\mathcal{L}$ be the the space of Borel-measurable functions $f:[0,1]\to[0,1]$, and $e:\mathcal{L}\times [0,1]\to[0,1]$ the evaluation given by $e(f,x)\mapsto f(x)$, then there is no $\sigma$-algebra on $\mathcal{L}$ such that the evaluation is jointly measurable. The result is a consequence of the rather complicated classification result in R. Aumann, Borel Structures for Function Spaces, Illinois Journal of Mathematics 5 (1961), pp. 614-630. Easier proofs of the main results can be found in the paper "Borel Structures for Function Spaces" (yes, same title) by B.V. Rao, Colloquium Mathematicum, 1971.

A $\sigma$-algebra on measurable functions I have actually seen used is the following: If $(S,\mathcal{S})$ and $(T,\mathcal{T})$ are measurable spaces, we endow the family of measurable functions between them with the $\sigma$-algebra generated by sets of the form $\{f:f(s)\in B\}$ with $s\in S$ and $B\in\mathcal{T}$. The author used this $\sigma$-algebra to show that to each Markov kernel from $S$ to $T$, there corresponds a certain probability measure on this $\sigma$-algebra. The paper is H. v. Weizsäcker Zur Gleichwertigkeit zweier Arten der Randomisierung, Manuscripta Mathematica 11 (1974).

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There is the Effros Borel structure. But that is a $\sigma$-algebra for the collection $F(S)$ of closed sets in a Polish space $S$. But it is again a standard Borel structure, so maybe it meets your criterion of "nice"...

Google "effros borel structure" for more information.

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Since I'm not that familiar with the literature having to do with Polish spaces and Borel structures, could you give me a few links? I'd really rather not spend hours wading through the google results, only some of which I have access to, trying to find an introduction written in such a way that someone who only has Rudin's Real and Complex Analysis under their belt might be able to understand (at least to the point of figuring out the basics of what's going on). –  Avi Steiner Aug 9 '12 at 1:13
    
You want Edgar to pore through Google hits to guess at what you might understand so that YOU don't have to spend hours going checking out links? I vote to close this thread. –  Bill Johnson Aug 9 '12 at 12:51
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I suppose start with Christensen's book. Probably in your library. It is supposed to start at a relatively elementary level. amazon.com/Topology-Borel-structure-applications-North-Holland/… –  Gerald Edgar Aug 9 '12 at 17:05
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@Gerald, @Bill I'm really sorry. I guess I came across a bit harsh. I should have just asked for a reference instead of blabbering on about what I do/don't already know. –  Avi Steiner Aug 10 '12 at 14:35
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