Most probability measures 𝒫 (and all measures that are continuous on their parameters) give zero probability to H0.

And some don't.

So (in Bayesian terms) H0 is a priori wrong w.p. 1.

Or not.

And in non-Bayesian terms, you’re calculating the likelihood of your measurements under two competing hypotheses, one of which is correct w.p. 0 even conditioned on one of the two hypotheses being correct.

If one of the two (H0 or H1) is correct and you don't know which one then it could be either... Of course is you knew a priori which one is correct you wouldn't be considering the other at all.