Support of a continuous distribution

Question

Suppose I have a continuous random variable $X$ on $\mathcal{R}^1$, with CDF $F(\cdot)$ and pdf $f(\cdot)$.

My understanding is that there are three equivalent definitions of the support of the random variable.

1. $S=\{x :\Pr(X\in B(x,r))>0$ for all $r>0\}$, where $B(x,r)$ is the interval $(x-r,x+r)$.
2. The smallest closed set $S$ such that $\Pr(X\in S)=1$.
3. The closure of $\{x:f(x)>0\}$.

I have two questions about this.

Question 1: Is it true that these three definitions are indeed equivalent?

Question 2: Consider the following, $S_0\equiv S\cap\{x:f(x)=0\}$. That is, $S_0$ is the subset of the support at which $f(x)=0$. Is it true that $S_0$ has Lebesgue measure 0?

Answer 1

The most general definition that I am aware of takes support of any Borel measure $\mu$ on a topological space $X$ to be the the smallest closed set $A$ such that $\mu(X\setminus A) = 0$ (cf. $\rm [I]). $

$\rm [II]$ rather elaborates the characterization (relevant to what OP is seeking through the three definitions) well:

Theorem 2.1. Let $X$ be a separable metric space and $\mu$ a (probability) measure in $X.$ Then there exists a unique closed set $C_\mu,$ satisfying $\rm{(i)}~ \mu(C_\mu) = 1,~\rm{ (ii)}$ if $D$ is any closed set such that $\mu(D) = 1,$ then $C_\mu\subseteq D.$ Moreover, $C_\mu$ is the set of all points $x\in X$ having the property that $\mu(U) > 0$ for each open set $U$ containing $x.$

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References:

$\rm [I]$ Real Analysis and Probability, R.M. Dudley, $2004, $ p. $227.$

$\rm [II]$ Probability Measures on Metric Spaces, K. R. Parthasarathy, Academic Press, $1967,$ sec. $2.2, $ pp. $27-28.$