Another definition for the support of a random variable

Question

Let $S_X$ be the support of a random variable $X$, \begin{equation} S_X = \{x \in \mathbb{R} : F(x + \varepsilon) - F(x - \varepsilon) > 0 \mbox{ for all } \varepsilon > 0\}\mbox{,} \end{equation} being $F : \mathbb{R} \to [0 , 1]$ the distribution function associated with $X$. Defining the family \begin{equation} \mathcal{C} = \{C \subset \mathbb{R} : C \mbox{ is a closed set and } P_X(C) = 1\}\mbox{,} \end{equation} where $P_X$ is the probability induced by $X$, I want to show that \begin{equation} S_X = \bigcap_{C \in \mathcal{C}} C\mbox{.} \end{equation} I have already proved that $S_X$ is a closed set in $\mathbb{R}$, so I have only to show that $S_X \subset C$ for all $C \in \mathcal{C}$, though my attemps are not successful. Could you give me a hint? Thank you very much in advance.

Answer 1

Hint: It will suffice to assume $\exists C \in \mathcal{C}, x \in S_X$ s.t. $x \notin C$ and derive a contradiction. Since $C$ is closed, there is some $\varepsilon>0$ s.t. the interval $(x-\varepsilon,x+\varepsilon)\notin C$.