TITLE: Could anybody give me an example of finite measure which has non-zero value for all non-empty measurable set?
QUESTION [5 upvotes]: I'm trying to construct a finite measure defined on Borel sigma algebra of $[0,1]$ such that every nonempty set has nonzero measure.
Can anybody give me a help to find out such measure?
Thanks in advance.

REPLY [18 votes]: Such a measure doesn't exist.
Assume that $\mu([0, 1]) = M < \infty$. Then the cardinality of $T_n = \{x \in [0, 1] \mid \mu(\{x\}) > \frac{1}{n}\}$ is at most $Mn$ and therefore finite. Since every nonempty set is supposed to have a positive measure, the identity $[0, 1] = \bigcup \limits_{n \in \mathbb{N}} T_n$ holds. But this is impossible, since $[0, 1]$ is uncountable and the countable union of finite sets is at most countable.