Reason for defining a quantity with "inf"

Question

In some applications (in my case statistics) I find quantities defined using "inf", e.g.

$ ABC = \inf\{x|F_X(x)\ge\alpha\}$

Why not define simply:

$F_X(x=ABC) = \alpha$

I imagine it has something to do with the possibility of multiple values for $x$ satisfying the condidion, right?

Assuming, single valued solutions to the condition could be garanteed, is there a prettier notation for defining ABC without using set builder notation? (Since there is only a single value, set notation seems non-intuitive.)

$ ABC = \{x|F_X(x)=\alpha\}$

Answer 1

The reason why one writes "inf" in this case is that there may not be any x satisfiying $F_X(x) = \alpha$. In this case the "equals" condition would yield an empty set and be useless. Therefore you need to define it as the infimum.

Consider for example $\alpha=0$, $x \in M = \{\frac{1}{n} | n \in \mathbb{N}\}$ and $F(x)=x$. Then there exists no $x$ such that $F(x) = \alpha$, but $\inf_{x \in M} \{x | F(x)\geq \alpha \} = \{0\}$.

If it can be shown that $\exists ! x \text{ }$ such that $F_X(x)=\alpha $, then you can simply define $x =F^{-1}_X(\alpha) $.

Answer 2

Your notation really doesn't make sense. Perhaps you can clarify it?

Note that the quantity $\{x:F_X(x)\geq \alpha\}$ is frequently written as $$F_X^{-1}([\alpha,\infty))$$ so the quantity in question is just $$\inf F_X^{-1}([\alpha,\infty))$$ The infimum is used because the set may not contain its lower bound (if it does contain this lower bound, then the infimum is simply the minimum). Except for that, you are essentially finding the "smallest" value of $x$ such that $F_X(x)\geq\alpha$. If there is a "hole" in the graph of $F_X$ at height $\alpha$, this would be the greatest value of $x$ that isn't mapped above the hole.