Why is the measure problem an issue if the universe is always finite? As a follow-up from this question, I was wondering how the measure problem applies to the idea of eternal inflation.
First, why is it even a problem? As time progresses, more universes are created in an exponential fashion. However, no matter how long you wait, the number of universes will always be finite. Why not just assign probabilities taking all universes into account, if there are only a finite number of them
For example, as Alan Guth was quoted:

In a single universe, cows born with two heads are rarer than cows born with one head. [But in an infinitely branching multiverse] there are an infinite number of one-headed cows and an infinite number of two-headed cows. What happens to the ratio?

Except, there are never going to be an infinite number of either type of cows; the number will always be finite, and I'd expect that the one-headed cows are simply more numerous than the two-headed cows.
Second, why do physicists not simply accept that probability has no meaning in the multiverse? Probability, for sure, is very useful in physics, but why feel the need to accept it as an immutable and proven law? For a long time, philosophers have questioned how exactly probability should be taken, and whether or not it is a valid tool to use. For example, ideas like the problem of induction (potentially) show us that the scientific method is not completely justified - we just use it in a pragmatic way. Why not accept the same with probability?
 A: 
As time progresses, more universes are created in an exponential fashion.  However, no matter how long you wait, the number of universes will always be finite. Why not just assign probabilities taking all universes into account, if there are only a finite number of them? 

The problem is using phrases like "as time progresses" or "how long you wait" carelessly. 
In relativity, time ordering can differ between observers -- for spacelike separate events, one cannot unambiguously say whether one came before, after, or at the same time as the other one. And all the universes are spacelike separated, essentially by definition. So saying "just count the universes at some time" has no content; you can define your notion of time to give essentially any answer at all.
Now, you might say that at the very least, there exist time cutoffs that give finite answers, so what's the problem? The problem is that the predictions depend very strongly on what kind of cutoff you use, and worse, the simplest and most sensible-sounding cutoffs lead to very strange results, such as the youngness paradox or an abrupt, uncaused end of time. As far as I'm aware, nobody has come up with an unambiguous, computable, simple rule that gives reasonable physical results.

Second, why do physicists not simply accept that probability has no meaning in the multiverse? 

This depends on who you ask, but if you ask me, the point of a scientific theory is to explain observations. And whether or not it achieves that depends on whether it can account for observed data -- which always is quantified using a notion of probability. If you lose this, you lose the ability to evaluate whether a theory is good or not.
