Which implications are true (if any) for a measure $\mu$:
- $\sigma$- finite $\implies$ locally finite
- locally finite $\implies$ $\sigma$-finite
My guess would be that both are false, but no counterexamples come to mind, since I am only used to working with $\sigma-$finite measure spaces, hence do not know many (if any) measures which do not satisfy that condition.
I was thinking about this because I was wondering why local finiteness is used in the definition of Radon measures instead of $\sigma$-finiteness, and I figured that the counterexamples to the above question might give me my answer.
Specifically, if there is a counterexample for the second implication which is a Radon measure, that would be greatly preferred.