Support of a Measure is Closed

Last updated Nov 1, 2022

Let $(X, \mathcal{B}(X), \mu)$ be a Borel Measure Space with Borel Measure $\mu$. Then $\text{supp}(\mu)$ is Closed.

Recall that Complement of Support of Measure is Union of all Null Open Sets and is thus Open by definition of Topological Space. Then, by definition of Closed, $\text{supp}(\mu)$ is Closed. $\blacksquare$