Notation for probability distribution (capital P) and density function (lowercase p)

Question

I'm confused by the differences in the notation used to denote probability distribution and the density function. My understanding is that the probability distribution is usually denoted by the capital letter $P$, while the density function is denoted by lowercase $p$.

From a text I'm reading, an application of Bayes's rule is as follows:

$$p(\alpha,\beta\mid C)\propto P(C\mid \alpha,\beta)p(\alpha)p(\beta)$$

In this case, why is $P(C\mid\alpha,\beta)$ in capital $P$? Shouldn't it all be lowercase $p$ throughout the right-hand side since the term on the left-hand side is a density function?

Answer 1

$P(C\vert\alpha,\beta)$ is not a density function, it's the probability of the event (or measurable set) $C$ conditioned by the random variables $\alpha$ and $\beta$.

Answer 2

Even if the book/context uses a different P when there's a vertical line meaning "given", certainly the two Ps with vertical lines should be written the same. And all P's here can be be read "the probability of".

As an aside, "Density function" is sort of ambiguous (cumulative density function? Probability density function only in the continuous case? Probability density function even in the discrete case where it's called probability mass function?). I wonder if the book(s)/lecture notes are actually using all p's the same at this point.