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    • CommentRowNumber1.
    • CommentAuthorStephenGaito
    • CommentTimeNov 18th 2009

    I want to do point-less measure theory. It is reasonably obvious (says he) that we can define measures solely on "measure-Frames"/"measure-Locales" as sigma-algebraic order-categories (or their opposites).

    Does anyone know the correct names for "measure-Frames"/"measure-Locales"? (If not suggestions would be welcome).

    I expect that the (correct?) definition of a generalized function should "obviously" live in such a "measure-Frames"/"measure-Locales" setup.

    Does anyone know of anyone, any-hints, or any-leads where I could find this theory already worked out?

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeNov 18th 2009
    • (edited Nov 18th 2009)

    If I understand correctly what you mean by ‘measure-Frames’, the correct term is boolean σ-algebra. (Just combine σ-algebra with boolean algebra.)

    Most of the ‘pointless’ measure theory that I've seen, however, does not start here but instead starts with something intended to be interpreted as the algebra of measurable real-valued functions on the space.

    • CommentRowNumber3.
    • CommentAuthorStephenGaito
    • CommentTimeNov 18th 2009

    Toby, many thanks for this.

    Yes I guess the best term for the underlying objects is a boolean ?-algebra. However I will also want to stress the Frame/Locale nature so I guess the best names when I want to do that is to simply call them "boolean ?-algebraic frame/locales".

    I can see that, when the dust settles, it will be easy to relate the current approach, via algebras of measurable real-valued functions on a space, to my frame/locale approach. I am, however, convinced that I do need the frame/locale approach.

    When the dust settles there should be a chain of "naturally" related structures:

    • an (appropriately defined) category,
    • is naturally related to an (appropriately defined) groupoid related to the category,
    • is naturally related to a frame/locale of "words" over the category,
    • is naturally related to a boolean ?-algebra frame/locale of "words" over the category,
    • is naturally related to mutually-dual pair of algebras of generalized functions and measures on the frame/locale of "words" over the category,
    • is naturally related to a "space" of Markov operators on the frame/locale of "words" over the category (using a form of the Schwartz kernel theorem)

    My penultimate goal is to, using these mutually-dual pair of algebras of functions/measures together with the space of Markov operators, study the dynamical system flow of functions/measures induced by a Markov operator. (Note that the invariant measures are interesting in that they provide useful "structure", however, "reality" is the non-invariant flow).

    My ultimate goal is to see this as a (reversibly) natural chain of relationships inside an encompassing category of (individual) categories. It is this overall structure which tells me how the different objects are related as the underlying (individual) category changes inside the encompassing category (i.e. as I take a limit of the (individual) categories in the encompassing category).

    So I guess the essential answer to my question is: "no-one has done this in this particular way yet"... and that I will have to add this work to my stack of to-do items.

    If anyone knows of anyone who has done this, or is interested in doing this, I would be very happy to simply quote their results ;-)

    • CommentRowNumber4.
    • CommentAuthorSM
    • CommentTimeMay 20th 2020
    • (edited May 20th 2020)
    See Olivier Leroy and Alex Simpson, maybe?
    https://www.sciencedirect.com/science/article/pii/S0168007211001874
    https://hal.archives-ouvertes.fr/hal-00741126/document
    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 22nd 2020
    • (edited May 22nd 2020)

    It may be 11 years late, but my recent paper https://arxiv.org/abs/2005.05284 explores the category of measurable locales. With the exception of their discrete parts, measurable locales have no points at all!