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?