TITLE: Bounded sets have finite measure
QUESTION [5 upvotes]: Is it in general true that bounded sets have finite measure?
I would like to use this result for an exercise.

REPLY [5 votes]: As a rather extreme counter example let $X=R^n$ and set $\mu(A)=+\infty$ if $A\subset X$ and $A$ is nonempty and $\mu(\emptyset)=0$, then $\mu$ is a measure which assigns infinite measure to all nonempty subsets (in particular the bounded ones). 
On the other hand, the lebesgue measure is finite on all bounded subsets. The amount of work required to show this depends on your definition of a lebesgue measurable set.