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Tim Hosgood (Apr 15 2021 at 15:06):Abstract:
In point-free (or abstract) measure theory measurable spaces are replaced by -complete Boolean algebras, measurable functions by Boolean homomorphisms, and measure spaces by measure algebras. This more general perspective has some advantages over the traditional pointwise approach to measure theory. For example, it facilitates the use of topos-theoretic techniques to study measurability. To this effect, a translation process between the internal language/ logic of certain Boolean topoi and the "usual" external language/ logic is required which we can accomplish by using the formalism of conditional analysis. We illustrate this with some recent applications in ergodic theory.
Tim Hosgood (Apr 15 2021 at 15:07):YouTube: https://www.youtube.com/watch?v=d94jIahj2JQ
Zoom: https://topos-institute.zoom.us/j/5344862882?pwd=Znh3UlUrek41T3RLQXJVRVNkM3Ewdz09
Tim Hosgood (Apr 15 2021 at 15:07):this talk will take place at 17:00 UTC today (i.e. in just under two hours from now)
Paolo Perrone (Apr 15 2021 at 21:14):Great talk!
Paolo Perrone (Apr 15 2021 at 21:25):A question for everyone, just out of my ignorance: is there a structural reason why measure algebras are required to be nonzero outside the bottom element, while usual measures allow null sets?
Matteo Capucci (he/him) (Apr 17 2021 at 06:10):IIRC what's the definition of measure algebra, the idea is to be the 'formal' analogue of what you get by quotienting a sigma-algebra with the ideal of null sets of a measure on it. Therefore the resulting valuation on the quotient is 0 only on the bottom element.
John Baez (Apr 17 2021 at 13:58):Hmm, requiring an inequality in a definition tends to cause trouble, so I'd expect the category of measure algebras to be a bit less nice than if we dropped the requirement that the measure of everything other than the bottom element be nonzero.... even if the examples we're mainly interested in have that property.
Morgan Rogers (he/him) (Apr 20 2021 at 16:53):I finally found time to watch this. Is there a way to reach Asgar to discuss this? I would like to help him with the question he asked at the end, but I would need to see more examples.
Valeria de Paiva (Apr 20 2021 at 16:55):@_Morgan Rogers (he/him)|277473 said:
I finally found time to watch this. Is there a way to reach Asgar to discuss this? I would like to help him with the question he asked at the end, but I would need to see more examples.
jasgar (at) math.ucla.edu
Morgan Rogers (he/him) (Apr 20 2021 at 17:51):Thank you!