Is there a reliable reference that shows that Bayes theorem holds for measures, densities, masses, or combinations of these?

As announced in the comment, I'd cite Theorem 1.31 in this book. It says:

Suppose that $X$ has a parametric family $mathcal{P}_0$ of distributions with parameter space $Omega$. Suppose that $P_thetall nu$ for all $thetainOmega$, and let $f_{X|Theta}(x|theta)$ be the conditional density (with respect to v) of X given $Theta = theta$. Let $mu_Theta$ be the prior distribution of $Theta$. Let $mu_{Theta|X}(cdot|x)$ denote the conditional distribution of $Theta$ given $X = x$. Then $mu_{Theta|X}llmu_Theta$, a.s. with respect to the marginal of $X$, and the Radon-Nikodym derivative is $$ frac{dmu_{Theta|X}}{dmu_Theta}(theta|x)=frac{f_{X|Theta}(x|theta)}{int_Omega f_{X|Theta}(t|theta),dmu_Theta(t)} $$ for those $x$ such that the denominator is neither $0$ nor infinite. The prior predictive probability of the set of $x$ values such that the denominator is $0$ or infinite is $0$, hence the posterior can be defined arbitrarily for such $x$ values.

In your cases of interest $nu$ is the Lebesgue measure.

Bayes' Rule is of course just an application of the definition of conditional probability/ density. But if you'd like something to refer to, it is stated for general distributions (i.e., encompassing both "densities" and "mass functions") in the introduction (see p. 7 of 2nd edition) of Bayesian Data Analysis by Gelman, Carlin, Stern, and Rubin.