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

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?

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