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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 30th 2020

    Added the Radon-Nikodym theorem.

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 30th 2020

    “Truly continuous” is a synonym for “absolutely continuous”?

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 30th 2020
    • (edited May 30th 2020)

    No. It is explained in the statment (“the latter means…”).

    Absolutely continuous is used in the corollary, which restricts to the σ-finite case.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 1st 2020

    Added an example when absolutely continuous does not imply truly continuous.

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorPaoloPerrone
    • CommentTimeJun 1st 2020

    I’m very pleased to see that so much measure theory is being added to the nLab. Thank you Dmitri!

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2020
    • (edited Jun 1st 2020)

    Yes, it’s great that Dmitri is adding content – here on measure theory and elsewhere!

    By the way, if you are fond of measure theory on the nnLab, please do consider updating the topic cluster list

    when you create a new entry, and don’t forget to add the floating topic cluster menu to all the entries on measure theory, by adding this code at the top of each entry:

      +-- {: .rightHandSide}
      +-- {: .toc .clickDown tabindex="0"}
      ### Context
      #### Measure and probability theory
      +-- {: .hide}
      [[!include measure theory - contents]]

    (If nothing else, this helps Google spread the word about the content you are adding.)

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 1st 2020

    Added the Lebesgue decomposition theorem.

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorPaoloPerrone
    • CommentTimeJun 2nd 2020
    • (edited Jun 2nd 2020)

    Actually, now that we are at it, I would like to ask if we can split the “measure and probability theory” context into two, namely, measure theory and probability theory. This for a variety of reasons:

    • While related, they are not quite the same subject, nor the same community (think algebraic topology and homotopy theory);
    • The nLab is getting enough articles in those topics to have enough to say about both separately;
    • There are many current categorical approaches to probability theory (such as Markov categories) which do not involve measure theory directly;
    • Conversely, there are many categorical approaches to measure theory (such as the theory of valuations) which, albeit one can use them indirectly to do probability theory, are primarily about measure theory itself.

    What does the community think?

    (Also, is this doable without manually editing a lot of pages?)

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 3rd 2020

    I agree, and I can add that probability theory studies random variables and their properties such as independence, as opposed to measure spaces.

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 3rd 2020

    Previously, one could see a list of backreferences indicating articles linking to the given article.

    I can no longer find it.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 3rd 2020
    • (edited Jun 3rd 2020)

    Re #8, I can see that one would want to distinguish the two areas, but at the moment the context is calling measure theory - contents. This has a ’thermodynamics’ section too. If we’re only limiting contexts to narrowly defined fields, that section would have to go.

    Many of the contents pages have entries in neighbouring fields, e.g., combinatorics - contents has a section on graph theory.

    I guess it all depends on how people see the function of these contexts. I barely ever use them.

    I see there’s a contents of contents page listing contents pages from 2011-08-10, which could be explored (and updated).

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020

    What does the community think?

    Do what you find useful for yourself and for users of your pages.

    More critical than cosmetic issues is always the content itself. If you take care of compiling a nice comprehensive list of contents for measure theory and/or probability theory, then I can lend a hand with fine-tuning the formatting/layout.

    Also, is this doable without manually editing a lot of pages?

    No. But if you take care of adding the content I can find 8 minutes to take care of such cosmetic issues. I have practice in doing that :-)

    Previously, one could see a list of backreferences indicating articles linking to the given article. I can no longer find it.

    This functionality was deactivated long ago because it produced a huge worm of terms at the bottom of each page, blindly wrapping around lines. It would of course be useful to have that list of terms be displayed not by default on each page but on demand. If we want that, I guess we have to ask in “Bugs and feature requests.”

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 10th 2020

    Another example of absolutely continuous, but not truly continuous measure.

    diff, v8, current

    • CommentRowNumber14.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 11th 2020

    Truly continuous measures form a free module of rank 1.

    diff, v9, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 13th 2020

    Additional remarks about absolute and true continuity.

    diff, v10, current