I have some data and a model, e.g.,

$H_0: \xi \sim \mathcal N(\mu_1 > \mu_0, \sigma_0)$,

in other words, this is the similar to the situation with "unequal means, alternative=greater".

How can I calculate the likelihood of this model? Numerical integration could help, but often this integral is divergent (upper limit is infinity). I can put some prior on possible $\mu_1$ values, but it is another problem.