# Is this a typo in the definition of support of a random variable?

Good evening, I'm reading about the support of a random variable in my lecture notes:

Definition 2.5(src) (Support) Let $$X$$ be a real-valued random variable on $$(\Omega,\mathcal{A},\mathbb{P})$$. The support of $$X$$, denoted $$\mathit{Supp}(X)$$, is defined as follows: $$\mathit{Supp}(X) = \{x\in\mathbb{R}; \ \forall N_x, \ \mathbb{P}(X \in N_x) \neq 0 \}$$ where $$N_x$$ is an open neighborhood of $$x$$.

Definition 2.6(src) (Support: discrete and continuous case). Let $$X$$ be a real-valued random variable on $$(\Omega, \mathcal{A}, \mathbb{P})$$.

• If $$X$$ is discrete (see definition 2.9), then $$\mathit{Supp}(X) = \overline{\{ x \in \mathbb{R}; \ \mathbb{P}(X = x) \neq 0 \}}.$$

• If $$X$$ is absolutely continuous with respect to the Lebesgue measure (see definition-theorem 2.1) and $$f_X$$ is a p.d.f. of $$X$$, does not have any isolated point, then $$\mathit{Supp}(X) = \overline{\{ x \in \mathbb{R}; \ f(x) \neq 0 \}},$$ where $$f$$ is a density of $$X$$.

Whereas the definition of support is given by Wikipedia's page as follows:

In practice, support of a discrete random variable $$X$$ is often defined as the set $$R_{X} = \{ x \in \mathbb{R} : P(X=x) > 0 \}.$$ And support of a continuous random variable $$X$$ is defined as the set $$R_{X} = \{ x \in \mathbb{R} : f_{X}(x) > 0 \},$$ where $$f_{X}(x)$$ is a probability density function of $$X$$.(src)

Clearly, the support in my lecture note is the closure of that from Wikipedia.

My question: Is it a typo in my lecture note?

Thank you so much for your clarification!

• When it comes to the support of a function, it is actually a matter of preference, as I have seen both practice (taking closure or leaving it as it is). But the support of measure, defined as in Definition 2.5, is always a closed set regardless of the lack of closure operation. So it makes sense to have closure operations in Definition 2.6. – Sangchul Lee Sep 27 '19 at 20:28
• I'm happy to see you again @SangchulLee :) Could you write your comment as an answer so that I can accept it? – LAD Sep 27 '19 at 20:32
• @SangchulLee I'm quite surprised that you have retyped the images in 3 of my questions in LaTex. This job takes much time. I wonder why you spent time doing so ^o^ – LAD Sep 27 '19 at 20:33
• No problem, I am just out of my PC now, so I will try to make an answer when I go back. Anyway, if you get used to $\LaTeX$, it actually takes not much time to write in it. For instance, it took me less than 5 minutes to do so. And why I am doing so is to make your posting better exposed to search engines, so other people have better access to it. I also encourage you to do so whenever available :) – Sangchul Lee Sep 27 '19 at 20:38
• @SangchulLee I really appreciate your kindness and contribution to the community. Honestly, I'm having a high workload with study, so I'm able to type LaTex when the image is not a True-HD PDF ;( – LAD Sep 27 '19 at 20:40