Define "Almost Always Less Than" Is there a formal definition of "almost always less than" or "almost always greater than"? I think one could define it using probabilities but not sure how to go about it. If one could show the following, then I think you could say $X$ is almost always less than a value $x$. Is there other ways of going about this?
$$
P(X<x)=1
$$
 A: The precise meaning of "almost always" depends on context. In measure theory it means "except on a set of measure $0$".
Probability is often discussed in a measure theoretic context. In that context your probability assertion makes sense. For example, on the unit interval the value of $x^2$ is almost always less than $1$ when $x$ is chosen at random from the uniform distribution since the set on which it has its maximum value $1$ is the singleton $\{1\}$ with measure (probability) $0$.
A: This means that
$$\mathbb{P}[X\geq x]=0$$
Thus there can be some sets where $X\geq x$ but they are all sets with probability zero
A: If something holds almost surely, it means it is true on a set of probability measure 1. Let $X$ be a r.v. on some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $c\in\mathbb{R}$. Then the following are by definition the same:
i) $X<c$ $\mathbb{P}$-almost surely
ii) $\{\omega\in\Omega: X(\omega)<c\}$ occurs with probability 1
iii) $\mathbb{P}(X<c)=1$
A: Yes, a more typical terminology would be to say “almost surely (a.s.) less than value”
$$X < c\;a.s \iff P(X<c)=1$$
Now, if the above holds and $\{\omega \in \Omega:X(\omega)\geq c\} = \emptyset$ then we can say “surely” since there isn’t even a set of probability $0$ that results in $X\geq c$
