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Categorical Counterpart to Relation bet Metric and Measure S

  1. Aug 23, 2015 #1

    WWGD

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    Hi, just curious. Sorry I am trying to get a handle on this , will try to make it more precise:
    I am trying to see if the following has a categorical parallel/counterpart.
    Consider the case of measure spaces (X,S,m) : X any space, S a sigma algebra, m a measure and that
    of metric spaces (Y,d) with ( I would say) continuous maps. Given a metric space (Y,d) , there always exists a measure space associated with it, given by the sigma algebra generated by the open sets ( themselves generated by open metric balls ), and measure is n-dimensional volume. But, a given measure triple (X,S,m) does not necessarily correspond to a metric space (X,d), i.e, (X,S,m) is not necessarily a measure triple associated to (X,d).
    (Phew!) Is there a way of expressing/describing the above using Category Theory, e.g., can we describe the above in terms of the non-existence of functors between the categories (Metric spaces, Cont. Maps) and (measure spaces with measurable maps)?
    Thanks, sorry for the rambling.
     
  2. jcsd
  3. Aug 28, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Aug 28, 2015 #3

    WWGD

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    Sure Greg, please give me some time.
     
  5. Aug 29, 2015 #4

    micromass

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    How would you define this measure? What does dimension mean here?
     
  6. Aug 29, 2015 #5

    WWGD

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    Ah, yes, sorry, I was thinking of ## \mathbb R^n ##. Still, the issue is whether the possibility/issue of an assignment of a measure/ sigma algebra to a metric space (the one generated by the open sets and n-dimensional voume) can be described functorially or has a parallel description in terms of category theory. So that, e.g., the existence of a functor between these two categories would give a yes answer and a nonexistence woud say no.
    Re the general issue,. maybe in general metric spaces we can use the Hausdorff measure and the Borel sigma algebra? I think this last may not work, I have not thought it through enough.
     
  7. Aug 29, 2015 #6

    micromass

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    So indeed, you can have the category ##\textbf{Top}## of topological spaces and the category ##\textbf{Meas}## of measurable spaces (sets with ##\sigma##-algebra). Then you indeed have a functor ##F:\textbf{Top}\rightarrow \textbf{Meas}##.

    There are of course various functors from ##\textbf{Meas}## to ##\textbf{Top}##, but you will probably be interesting in some kind of left-or right adjoint of ##F##.
     
  8. Aug 29, 2015 #7

    WWGD

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    Ah, thanks, been out of school for too long.
     
  9. Sep 2, 2015 #8

    gill1109

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    Given a metric space one can associate many different sigma algebras with it. For instance: the sigma algebra generated by the open sets; the sigma algebra generated by the open balls; and so on. I don't know of any obvious way to define a measure on either of these sigma algebras given only the metric. There is of course a whole theory of Hausdorff dimension and Hausdorff measure. So I think your assumptions are not true and hence your question ill-posed.
     
  10. Sep 2, 2015 #9

    WWGD

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    Please refer to my post #5 where I made my argument more specific, refined. I did refer to the general issue of the _possibility_ of this correspondence , and I also referred to Hausdorff metric and Hausdorff measure.
     
  11. Sep 4, 2015 #10

    gill1109

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    Thanks. We need an expert on category theory to answer this.
     
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