# Any kind of result giving a sufficient condition for when a measure arises from the Riesz representation theorem?

Is there any sort of result known that gives a set of conditions on a measure space which are sufficient for it to be such that it arises from a linear functional on a locally compact Hausdorff space via the Riesz representation theorem? Or more generally is there a good place for me to look for work on classification of measure spaces?

• I guess for starters, you need some condition for a measurable space to be the Borel $\sigma$-algebra of some LCH topology. I would be interested in that in itself. – Nate Eldredge Feb 14 at 16:46
• The "bible" of measure theory is Fremlin's book. It's dense but you could look there. – Nate Eldredge Feb 14 at 16:52
• @PiotrHajlasz I think the question is about the "if you have a measure on a LCH space" part. How do you know that you are in that situation? If I were to give you any measure space, how would you decide whether or not the sigma algebra is the borel-sets of some LCH space? And can you decide it in a way that gives you enough information to see if my measure is Radon or not? – Johannes Hahn Feb 14 at 19:46
• @PiotrHajlasz How do get $C_0(X)$, if you only know $(X,\Sigma,\mu)$ ? In particular: How do you get the topology from the measure? In general you can't, since the topology is not uniquely determined from the measure space and there are measure spaces that do not come from a topological space at all. So how do you decide whether you're in such a case? And if you're not, how do you decide if one of the possible compatible topologies is LCH ? – Johannes Hahn Feb 14 at 20:10
• @JohannesHahn OK. I misunderstood the question. I was not a careful reader. I am deleting my stupid comments. – Piotr Hajlasz Feb 14 at 21:01