# “Nice” sigma-algebra on set of measurable functions

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

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"...