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Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space.

Based on

  1. Discrete Random Variables May Have Uncountable Images
  2. Understanding definition of a continuous random variable
  3. https://en.wikipedia.org/wiki/Support_(mathematics)#In_probability_and_measure_theory
  4. Is this a typo in the definition of support of a random variable?

( I'm trying to not use words like 'open' and 'closed' to reduce as much measure theory & topology as possible. )

About the 'in practice' part.

Q1

If $X$ is discrete with probability distribution function $f_X$, then are the following definitions of support $\text{supp}(X)$ the same?

  1. $\{x \in \mathbb R | f_X(x) > 0\}$, where $f_X(x) = P(X=x)$
  2. Smallest $R_X$ with $P(X \in R_X)=1$
  3. Smallest closed $R_X$ with $P(X \in R_X)=1$

Q2

If $X$ is continuous with probability distribution function $f_X$, then are the following definitions of support $\text{supp}(X)$ the same?

  1. $\{x \in \mathbb R | f_X(x) > 0\}$
  2. Smallest $R_X$ with $P(X \in R_X)=1$
  3. Smallest closed $R_X$ with $P(X \in R_X)=1$
BCLC
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    What do you think? – Henry Aug 27 '24 at 19:15
  • from your Wikipedia link "In probability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution." – Henry Aug 27 '24 at 19:26
  • For the continuous case, the pdf is not unique. You can pick an single $x_0$ with $f(x_0)>0$ and set it to $f(x_0)=0$. The distribution remains unchanged. So term "closed" cannot be formally dropped. But "in practice", it doesn't matter, because we don't care any null set with probability zero that never changes probability statements. – Zack Fisher Aug 27 '24 at 19:27
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    Also, "in practice", I don't think we care about the support of a random variable; the support of the distribution is of practical interest. – Zack Fisher Aug 27 '24 at 19:29
  • @ZackFisher Defining random variables in terms of support, not image – BCLC Aug 27 '24 at 19:41
  • @ZackFisher well in this case 'support of $X$' refers to 'support of $f_X$' (if any such $f_X$ exists) ? – BCLC Aug 27 '24 at 19:43
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    For Q1, 2 and 3 seem not equivalent in general. We can define a pmf $p$ over $\mathbb{Q}$ only by choosing an enumeration of the rationals $(q_n)_n$ and setting $p(q_n)=2^{-n}$. Then the smallest set with probability one is $\mathbb{Q}$ but the smallest closed set with probability one would be $\overline{\mathbb{Q}}=\mathbb{R}$. – Snoop Aug 27 '24 at 19:56

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