TITLE: Why is the support of a measure closed?
QUESTION [5 upvotes]: I found the following statement on Wikipedia

The support of a measure is closed in $X$ as its complement is the union of the open sets of measure 0.

where $X$ is from a topological space $(X, T)$ and the mentioned measure, call it $\mu$, is on the space $(X, \mathcal{B}(T))$. 
Why are the sets of measure zero open?

REPLY [7 votes]: It does not mean that all sets with measure $0$ are open, but that the complement of the support is the union of open sets with measure $0$.
See the definition of the support:

the support (or spectrum) of $μ$ is defined as the set of all points $x$
  in $X$ for which every open neighbourhood $N_x$ of $x$ has positive measure

This means that if $x\not\in supp (\mu)$, there exists $N_x$ an open neighborhood of $x$ such that $\mu(N_x)=0$. Hence the property.