Support of a finite Borel measure on the plane

Let $\mu$ be a finite positive Borel measure on $\mathbb{C}$ (Radon measure on an LCH space). Call an open set $V\subseteq \mathbb{C}$ $\mu$-null when $\mu (V)=0$. The support of $\mu$ is the complement of the union of all open $\mu$-null sets,

$\mathrm{supp}\mu :=\mathbb{C}\setminus \bigcup \{V\subseteq \mathbb{C}:V\text{ open and }\mu (V)=0\},$

which is a closed subset of $\mathbb{C}$ (Interior, closure, boundary, limit point, isolated point and dense subset of a metric space). Equivalently, $x\in \mathrm{supp}\mu$ if and only if $\mu (V)>0$ for every open set $V\ni x$. A measure is carried by a Borel set $A$ when $\mu (\mathbb{C}\setminus A)=0$, the terminology of A positive, signed, or complex measure concentrated on a measurable set. We say $\mu$ has compact support when it is carried by some compact subset of $\mathbb{C}$.

The support is the smallest closed carrier, and the argument is choice-free. Let $\mathcal{U}$ be the union of all open $\mu$-null sets. The rational open squares $B$ of ${\mathbb{R}}^2\cong \mathbb{C}$ form a countable basis (${\mathbb{Q}}^n$ is a countable dense subset of ${\mathbb{R}}^n$, and rational open boxes form a countable basis). Let $\mathcal{N}$ be the countable family of those basis squares with $\mu (B)=0$. If $x\in \mathcal{U}$, then $x$ lies in some open $\mu$-null $V$, and the basis property supplies a square $B\in \mathcal{N}$ with $x\in B\subseteq V$; conversely every $B\in \mathcal{N}$ is contained in $\mathcal{U}$. Hence $\mathcal{U}$ is the countable union of the $\mu$-null sets $\mathcal{N}$ and is itself $\mu$-null by countable additivity (Measures on sigma-algebras). Therefore $\mu (\mathbb{C}\setminus \mathrm{supp}\mu )=0$: the support carries $\mu$. If $F$ is a closed set with $\mu (\mathbb{C}\setminus F)=0$, then $\mathbb{C}\setminus F$ is an open $\mu$-null set, so it is one of the sets in the defining union and $\mathrm{supp}\mu \subseteq F$; the support is thus the smallest closed set carrying $\mu$. In particular $\mu \mathrm{\ne }0$ if and only if $\mathrm{supp}\mu \mathrm{\ne }\mathrm{\varnothing }$, and if $\mu$ is carried by a compact $K$, then $\mathrm{supp}\mu \subseteq K$ is compact.