[ants_everywhere]

It's standard to set the null hypothesis to be a measure zero set (e.g. mu = 0 or mu1 = mu2). So the probability of the null hypothesis is 0 and the only question remaining is whether your measurement is good enough to detect that.

But even though you know the measurement can't be exactly 0.000 (with infinitely many decimal places) a priori, you don't know if your measurement is any good a priori or whether you're measuring the right thing.

[energy123]

The probability is only zero a.s., it's not zero. That's a very big difference. And hypothesis tests aren't estimating the probability of the null being true, they're estimating the probability of rejecting the null if the null was true.

[ants_everywhere]

It's less of a big difference than it might seem, because it takes infinitely long to specify a real number to infinite precision. If you think about something like trying to tell if you hit the exact center of the bullseye, you eventually get down to the quantum mechanical scale and you find that the idea of an atom being in the exact center isn't even that well defined.

In a finite or countable number of trials you won't see a measure zero event.

> they're estimating the probability of rejecting the null if the null was true.

Right, but the null hypothesis is usually false and so it's a weird thing to measure. It's a proxy for the real thing you want, which is the probability of your hypothesis being true given the data. These are just some of the reasons why many statisticians consider the tradition of null hypothesis testing to be a mistake.