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Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, complements do not always exist, however finite meets and countable/arbitrary joins do exist. Would there be any significant difficulty or difference in developing a measure theory for frames? Is existence of complements essential for measure theory?

Motivation

It is sometimes argued that open sets in topology (think of frames as pointless topology) capture the intuitive notion of observable events (e.g. see Steven Vickers' book "Topology via Logic"). The complement of an observable event does not need to be an observable event in general: e.g. consider "there exists a white crow", assume that the number of crows are practically infinite so one cannot check all of them and observe that this statement is false, however to affirm it we need to observe a single white crow. It seems natural to want to assign probability to observable events and have a more general notion of measures that doesn't need complement.

Apologies if my question is naive.

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    $\begingroup$ The category of measurable spaces and measurable maps happens to be a full subcategory of the category of locales, see ncatlab.org/nlab/show/measurable+locale. Presumably one could then define (say) Radon measures on arbitrary locales and relate them to measures on the corresponding measurable locales obtained by the relevant universal property, for example, see ncatlab.org/nlab/show/…. $\endgroup$ – Dmitri Pavlov Oct 24 '12 at 15:45
  • $\begingroup$ @Dmitri, thanks, I will look into the links. $\endgroup$ – Kaveh Oct 24 '12 at 20:17
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Hello,

I think the following paper (partially) addresses your question:

Alex Simpson, "Measure, Randomness and Sublocales". In Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642–1659

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In fact, the analogue of measures for frames has in fact been studied under the name "valuations." As far as I'm aware, concept and name originates from the following papers:

Claire Jones and Gordon Plotkin, "A probabilistic powerdomain of evaluations". 1989.

Claire Jones, "Probabilistic Non-determinism". 1989. PhD thesis.

In addition to the reference suggested in the other answer, I suggest the following references:

Steve Vickers, "A localic theory of lower and upper integrals". 2009.

Steve Vickers, "A monad of valuation locales". 2011.

Thierry Coquand and Bas Spitters, "Integrals and valuations". 2009.

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