Q: Classifying topological spaces by measures While looking at some spaces, I happend to know, that in some spaces (like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)!
some examples: in $\mathbb R^n$ a Null set is totally-non-connected, nowhere-dense seems to emply Nullnes, in some spaces every null has a empty interior,but in others not;and more...
so my question is:


*

*Are there any usefull functors from TOP to the "category of measures"(or whatever its called), like the simple  ones to GROUP?

*Where can i find some book on this topic?
