Defining measures over frames in place of $\sigma$-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:15:35Z http://mathoverflow.net/feeds/question/110512 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110512/defining-measures-over-frames-in-place-of-sigma-algebras Defining measures over frames in place of $\sigma$-algebras Kaveh 2012-10-24T05:46:30Z 2013-02-02T02:27:45Z <p>Normally <a href="http://en.wikipedia.org/wiki/Measure_%28mathematics%29" rel="nofollow">measures</a> and <a href="http://en.wikipedia.org/wiki/Probability_space" rel="nofollow">probability spaces</a> are defined over <a href="http://en.wikipedia.org/wiki/Sigma_algebra" rel="nofollow">$\sigma$-algebras</a>. I was wondering what would happen if one tries to define it over <a href="http://en.wikipedia.org/wiki/Frames_and_locales#Frames_and_locales" rel="nofollow">frames</a> in place of $\sigma$-algebras? (i.e. 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?</p> <h3>Motivation</h3> <p>It is sometime 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 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.</p> <p>Apologies if my question is naive.</p> http://mathoverflow.net/questions/110512/defining-measures-over-frames-in-place-of-sigma-algebras/110537#110537 Answer by Matteo Mio for Defining measures over frames in place of $\sigma$-algebras Matteo Mio 2012-10-24T12:59:35Z 2013-02-02T02:27:45Z <p>Hello,</p> <p>I think the following paper (partially) addresses your question:</p> <p><a href="http://homepages.inf.ed.ac.uk/als/Research/Sources/mrs.pdf" rel="nofollow">Alex Simpson, "Measure, Randomness and Sublocales".</a> In Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642–1659</p>