Please correct me if I am not correct. On a Borel sigma algebra on a topological space $S$,
- a locally finite measure is defined as a measure for which every point of the measure space has a open neighbourhood of finite measure.
- A local finite subset is defined as a subset with which the intersection of every compact subset is a finite set. (I think it is a pure topological concept?)
I wonder
- why is it that every locally finite counting measure on $S$ is of the form $\sum_{x\in \Lambda}\delta_x$ where $\Lambda$ is a locally finite subset of $S$?
- For $\sum_{x\in \Lambda}$ to be well-defined, is $\Lambda$ required to be countable, i.e. to avoid being uncountable?
- Is the converse true, i.e. is it true that a counting measure on $S$ is locally finite if and only if it is of the form $\sum_{x\in \Lambda}\delta_x$ where $\Lambda$ is a locally finite subset of $S$?
In general, is a counting measure on a measurable space $X$ defined as $\mu(B) := \#\{A \cap B\}$ for some fixed and arbitrary subset $A$ of $X$? This is how I infer from "locally finite counting measure", but Wikipedia says a counting measure is defined as the cardinality function of measurable subsets.
Is a measure counting measure if and only if it is of the form $\sum_{x\in A}\delta_x$ where $A$ is a fixed and arbitrary subset of $X$? (Similar to a previous question, does $A$ need to be countable for $\sum_{x\in A}$ to be well-defined?)
Thanks and regards!