Support of a Measure
# Definition 1
Let $(X, \mathcal{B}(X), \mu)$ be a Borel Measure Space with Borel Measure $\mu$. The support of $\mu$, denoted $\text{supp} (\mu)$, is defined $$\text{supp}(\mu) = {x \in X : \forall N_{x} \in \mathcal{N}{x}, \mu(N{x}) > 0}$$ where $\mathcal{N}_{x}$ denotes the collection of all Open sets containing $x$.
# Properties
- Complement of Support of Measure is Union of all Null Open Sets. This gives an alternate definition.
- Support of a Measure is Closed. This establishes that $\text{supp} (\mu) \in \mathcal{B}(X)$.