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I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets...

But at least as far as I can tell, it doesn't quite work - if $(X,\Sigma,\mu)$ is a measure space and $X$ is also given the topology $\Sigma$, then we do get a presheaf $M:O(X)\rightarrow\mathbb{R}_{\geq0}$, where $\mathbb{R}_{\geq0}$ is the poset of non-negative real numbers given the structure of a category, and $M(U) = \mu(U)$, and $M(U \subseteq V)$ = the unique map "$\geq$" from $\mu(V)$ to $\mu(U)$, but this is not a sheaf because for an open cover {$U_i$} of an open set $U$, $\mu(U)$ is in general not equal to sup $\mu(U_i)$ (and sup is the product in $\mathbb{R}$).

First off, is the above reasoning correct? I'm still quite a beginner in category theory, but I've seen on Wikipedia something about a sheafification functor - does applying that yield anything meaningful/interesting? Does anyone have good references for measure theory from a category theory perspective, or neat examples of how this perspective is helpful?

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4 Answers 4

up vote 13 down vote accepted

For a sheaf-theoretical interpretation of measure theory, measure spaces are the wrong objects, you want measure algebras and then consider certain Grothendieck topologies on a Boolean algebra.

For measure algebras, check out volume 3 of FRemlin's 5-volume opus dedicated to measure theory. For more sheaf-theoretical stuff, google for Mathew Jackson's phd thesis on measure theory in the context of topos theory.

Hope it helps, regards, G. Rodrigues

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Thanks for the amazing answer to that question :) –  CSA Oct 29 at 16:56

http://etd.library.pitt.edu/ETD/available/etd-04202006-065320/unrestricted/Matthew_Jackson_Thesis_2006.pdf

http://front.math.ucdavis.edu/0912.4914

R. Borger "FUbini Theorem from a Categorical Viewpoint" in "Category Thery at work" (H.Herrlich- H-E Prost (eds) " "Helderman Verlad Berlin" 1991

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Thanks for the links and references! –  Zev Chonoles May 28 '10 at 17:11

In Johnstone's book "Topos theory" there is an interesting example of a topos based on a measure space, see 6.62(ii). If $(X,\Sigma,\mu)$ is a measure space, then the site consists of the poset $\Sigma$, with a coverage generated by countable families of inclusions $(B_i \to B)$ such that $B - \cup B_i$ has measure 0.

The fascinating thing about this topos is that the Dedekind reals in this topos is the sheaf of random variables on $X$, i.e. equivalence classes of measurable real-valued functions modulo almost-everywhere equality.

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3  
You mean the things that everyone else calls measurable functions, no? –  Harry Gindi Nov 30 '09 at 21:30

In some sense you always have a natural topology that comes with a measure space. Here is what I mean by that.

Once you have a measure space you can associate an algebra of measurable sets with this space (you may want to factor this algebra by an ideal of sets o measure zero). This measure algebra, as any other Boolean algebra, has Stone Space associated to it. It is such a compact zero-dimensional topological space, that algebra of its clopen (closed and open) sets is isomorphic to the given Boolean algebra. You can then transfer measure to that Stone space to make it into a measure space (in general the measure will not be $\sigma$-additive, but you can recover it from the $\sigma$-additivity on the algebra).

This measure space is not isomorphic to the space we started from, but they have isomorphic measure algebras. Since in measure theory we care only about measure algebra we don't lose any information, but you now also have a topology on the space such that all clopen sets are measurable. In some natural cases all Baire sets will be measurable (for example, for Lebesgue spaces). So, it is probably more natural to consider sheaves on the Stone space of the measure algebra.

UPDATE: For completeness I decided to add some references. First of all there is a great (but in some places not easy to read) treatise by Fremlin Measure Theory. You can download all the volumes for free. Volume 3 contains a huge amount of information about measures on Boolean algebras. Another source is Vladimirov's book Boolean Algebras in Analysis (especially chapter 7, where he talks about Caratheodory construction on Boolean algebras).

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If we allow $\mu$ to take on infinite values, won't we run into some trouble trying to recover the $\sigma$-additivity? –  Harry Gindi Nov 30 '09 at 10:02
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OK, I had to be more precise. Measure, even if it can have infinite values, is always defined on clopen subsets of the Stone space. It is sigma additive in the following sense. If you have a disjoint sequence of clopen sets $a_i$ and if $a$ is the smallest clopen set that contains all $a_i$'s then measure of this $a$ is the sum of measures of $a_i$. This is just because of isomorphism between Boolean algebras. You can then, probably (I didn't check this), apply Caratheodory construction to get a measure on the Stone space. –  Konstantin Slutsky Nov 30 '09 at 10:37
    
Thanks for the helpful explanation and links - it sounds like this is the right way to go. –  Zev Chonoles Nov 30 '09 at 17:59

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