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?