= Solution
For a <Borel probability measure> $\mu$ on a <Polish space> $E$, its <support of a measure> is
$$
\boxed{\operatorname{supp}\mu=\{x\in E:\mu(U)>0\text{ for every open neighborhood }U\text{ of }x\}.}
$$
Equivalently, for a compatible metric, every open ball about $x$ has positive measure. The complement is open: any zero-measure open neighborhood of a point is a zero-measure neighborhood of all its points. Hence the support is closed.
A Polish space has a countable base. Its support complement is therefore a countable union of zero-measure basic open sets, so $\mu(\operatorname{supp}\mu)=1$. If $F$ is any closed full-measure set, a point outside $F$ has the zero-measure open neighborhood $E\setminus F$ and is outside the support. Thus the support is also the smallest closed set of full measure. Countability is what justifies the full-measure assertion; an arbitrary uncountable union of null sets would not suffice.
Back to article page
