So, given a topological space $S$ we can construct its Borel sigma-algebra $\mathcal{B}(S)$. Does it mean that we can construct a measure $\mu$ on this sigma-algebra as well? Say, discrete topology on the circle implies $\mathcal{B}(S) = 2^S$, and hence we know that we cannot construct a measure.
This leads to the idea that certainly we need some restrictions on the topological space. What are useful sufficient conditions?
I am especially interested in the conditions like
– separable;
– second countable;
– metrizable;
Best Answer
This question is an oldie, but I feel that it deserves a more elaborate answer, so here goes.
As you said, to every topological space $X$ one can associate the Borel $\sigma$-algebra $\mathcal{B}_X$, which is the $\sigma$-algebra generated by all open sets in $X$. Now $(X,\mathcal{B}_X)$ is a measurable space and it is desirable to find a natural Borel measure on it. By Borel measure I simply mean a measure defined on $\mathcal{B}_X$ and by "natural" I mean that it should be compatible with the topology of $X$ in some sense (otherwise, $X$ is just an abstract set). There are several compatibility conditions one can impose, which are motivated from the fact that the Euclidean spaces (with Lebesgue measure) satisfy all of them. The following are most often encountered:
In addition, there are some general measure-theoretical conditions one can impose on $\mu$ which ensure that it obeys to some general measure-theoretical theorems (e.g. Fubini). The most useful ones are:
Also, you probably want $\mu$ to be nondegenerate ($\mu \ne 0$).
The above conditions are certainly not mutually independent and even for the most general topological spaces there are some obvious implications. For instance:
In the highest generality (arbitrary topological spaces), there is almost nothing nontrivial one can say, and one may not be able to find any nice Borel measure on the space. Therefore one restricts to some nice class of topological spaces, in which an interesting theory can be developed. Among them we have:
Almost any space which shows up in applications belongs to one of these two classes. For instance, real (and p-adic?) Lie groups and their homogeneous spaces always belong to the first class.
Here are some references: for the theory of measures on locally compact topological spaces, there's a book called "Measure and Integration" by König. It is somewhat technical but very general. For Polish spaces, there's a very nice book by Parthasarathy called "Probability Measures on Metric Spaces". "Handbook of measure theory" probably discusses these subjects in length too.