Is "almost any $t$" valid mathematical terminology?
I am familiar with almost surely, almost everywhere, etc. Here is the situation in which I am interested. Suppose we have two (discrete time) stochastic processes $\{X_t\}$ and $\{Y_t\}$ (taking values on the whole of the real line), with time $t$ indexed by the non-negative integers. I want to state a result which says:
If $X_t \neq Y_t$, for almost any $t$, then [statement A].
To expand on the intended meaning: For statement A to be valid, it is required that $X_t \neq Y_t$ for an infinite number of times but for a finite number of times one may or does have $X_t$ equaling $Y_t$.
As an example, the statement A is valid if, for example, $X_5 = Y_5$, $X_6 = Y_6$, and $X_7 = Y_7$ but for all integers $s$ such that $s \geq 8$, one has $X_s \neq Y_s$, almost surely.
Is the statement ``If $X_t \neq Y_t$, for almost any $t$, then [statement A]" the best way to say it?
I am not a mathematician by background but I want it to be sound and professional if read by a top class mathematician.
I am not a big fan of saying "almost every $t$", for two reasons: it is a bit overkill to use measure theory and it is unclear w.r.t. what measure this would hold. More precisely, I do not even see what measure you could be using. The only natural measure on the set of natural numbers (or on the set of integers) is the counting measure. In this case, "almost every $t$" coincides with "every $t$", so it is not very useful to you.
Instead, I would turn to the language used for set-theoretic limits (see e.g. here for definitions). At least in the case of $\mathbb N$, one would say "for all but finitely many $t$" (if you look at the link above, there is even a [more or less] official shortcut "a.b.f.o." for "all but finitely often") or also that "some property $A_t$ holds eventually". This corresponds to the $\liminf$ which you might be even interested in using. For example (in the case $t\in\mathbb N$), saying that "$X_t \neq Y_t$ for all but finitely many with probability 1" is the same as $$ \mathbb P\left( \liminf_{t\to +\infty} \{ X_t \neq Y_t\} \right) = 1. $$ And note that, because $\mathbb N$ is countable, this is also the same as "$X_t\neq Y_t$ with probability 1 for all but finitely many $t$". This can naturally be also formulated with the complement, being equivalent to $$ \mathbb P\left( \limsup_{t\to+\infty} \{X_t = Y_t\} \right) = \mathbb P\left(\text{ $X_t = Y_t$ infinitely often }\right) = 0. $$ Here, the corresponding shortcut is "i.o.".
Now, if you are looking at a process on $\mathbb Z$, then it might be less elegant to use the set-theoretic limits, but maybe there is a way. But independently of that, you can still use expressions like a.b.f.o or i.o.