I want to show that the probability that a random variable $X$ is smaller than a set of variables $C=\{C_1,C_2,...\}$ is equal to $a$. I use notation $X<C$ to show that $X$ is smaller than all $C$s. But set $C$ may be empty. How should I handle situation $P(X<C)$ when $C$ is empty? What is the right notation for it? Can I write $P(X<\emptyset)=1$?
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What is the interpretation of $X<C$ to begin with? – YJT Apr 24 '21 at 04:58
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I use notation $X<C$ to show that $X$ is smaller than all $C$s – Optimized Life Apr 24 '21 at 05:01
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In that case, I would use $X<\inf C$ and then https://math.stackexchange.com/questions/432295/infimum-and-supremum-of-the-empty-set#:~:text=If%20we%20consider%20subsets%20of,the%20empty%20set%20is%20%E2%88%92%E2%88%9E. – YJT Apr 24 '21 at 05:11
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@YJT Thank you, can you explain in more detail? – Optimized Life Apr 24 '21 at 05:19
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Based on the clarification in the comments:
You compare $X$ to all the elements of $C$, and the event $X<C$ in your notation is actually $X<\inf C$ in the standard one. (you compare random variables with numbers, not sets.)
Thus, if $C=\emptyset$ there is no problem: $\inf \emptyset=\infty$ (explanation) and thus $\Pr(X<\inf \emptyset)=1$. If you want to keep your notation, I guess there is no problem writing $X<\emptyset$ but it should be clearly defined in your text as it is not standard nor common.
YJT
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1If $C={0,1}$ then $\inf C=0$ and $\sup C=1$. To be smaller than all the elements, it should be smaller than the $\inf$ (by definition - inf is the highest lower bound). – YJT Apr 24 '21 at 05:37