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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?

Ralph
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    The equivalence is a standard exercise in point-set topology intended to help you understand the definition of a closed set and the closure of a subset. For an analysis that sheds light on question 2, see my recent post (about the zeros of density functions) at https://stats.stackexchange.com/a/600802/919. However, that doesn't really answer the question. I believe a "fat Cantor set" can be used to construct counterexamples. – whuber Jan 05 '23 at 15:26
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    should (2) be "The smallest closed set $S$ such that $\Pr(X\in S)=1$" ? – Henry Jan 05 '23 at 16:00
  • Smallest, of course! – Ralph Jan 05 '23 at 16:12

1 Answers1

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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 $x.$

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. $27-28.$

User1865345
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