Categorical Counterpart to Relation bet Metric and Measure S

[WWGD]

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.

 

[micromass]

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.

How would you define this measure? What does dimension mean here?

 

[WWGD]

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.

 

[micromass]

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##.

 

[WWGD]

Ah, thanks, been out of school for too long.

 

[gill1109]

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.

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.

 

[WWGD]

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.

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.

 

[gill1109]

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.

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