It's not a homework, actually I was reading an article where the following was stated. Let $\Omega$ be a toplogical space and $\mathcal{F}$ its Borel $\sigma$-algebra, i.e. the $\sigma$-algebra generated by its open sets. Let $\mu$ be a probability measure on $\Omega$. The support of $\mu$, denoted by $\text{supp}\mu$, is a closed subset of $\Omega$ which can be defined in three equivalent ways:
(1) the set of all $\omega \in \Omega$ such that every neighborhood of $\omega$ has nonzero measure.
(2) the intersection of all closed sets of measure 1.
(3) the complement of the union of all open sets with measure zero.
The equivalence between (2) and (3) is obvious since if $A$ is an open set of $\Omega$ with measure zero its complement $A^{c}$ is closed and has measure 1 and vice-versa. However, I have no idea how to prove that (1) is equivalent to (2) or/and (3). How do I address this problem?
Best Answer
If $x\in \Omega$ has a neighbourhood of measure $0$, then $x$ is contained in the union of the negligible open sets, because it is contained in a (necessarily negligible) open set that lives inside the negligible neighbourhood; if $x$ is in the union of the negligible open sets, then $x$ has a neighbourhood of measure $0$: for instance, a negligible open set containing $x$.
Therefore $(1)\Leftrightarrow (3)$.