Complement of Support of Measure is Union of all Null Open Sets

Last updated Nov 1, 2022

Let $(X, \mathcal{B}(X), \mu)$ be a Borel Measure Space with Borel Measure $\mu$. Then $$\text{supp} (\mu)^{C} = {U \subset X : U \text{ is open}, \mu(U) = 0}$$

Denote $\mathcal{N}{x} := {U \subset X : U \text{ is a neighborhood of }x \in X}$ . By definition of Support of a Measure, we know that $$\text{supp}(\mu) = {x \in X : \forall U \in \mathcal{N}{x}, \mu(U) > 0}.$$ Then $$\begin{align*} \text{supp}(\mu)^{C} &= {x \in X : \exists U \in \mathcal{N}{x}, \mu(U) = 0}\\ &\subset \bigcup\limits {U \subset X : U \text{ is open}, \mu(U) = 0}\\ &= {x \in X : U \text{ is open}, \mu(U) = 0, x \in U}\\ &= {x \in X : U \in \mathcal{N}{x}, \mu(U) = 0}\\ &\subset {x \in X : \exists U \in \mathcal{N}_{x}, \mu(U) = 0}\\ &= \text{supp}(\mu)^{C}, \end{align*}$$ so $$\text{supp} (\mu)^{C} = {U \subset X : U \text{ is open}, \mu(U) = 0}.$$ $\blacksquare$

1. This gives an alternate definition for Support of a Measure

• Neighborhood of a Point