Bayesian Decision Theory: Likelihood Test Probability Notation

Question

I'm having trouble understanding the notation $p(y|H_1)$ in the following example:

What we have is a variable $Y$ that we're trying to estimate. Our hypotheses are $H_0$ and $H_1$ which both have apriori probabilities $P_0$ and $P_1$ respectively.

Now my book says that the relationship between the hypotheses and the observed quantity $Y$ is given in the form of a probabilistic "measurement model": $$P_{Y|H}(y|H_0)\quad \& \quad P_{Y|H}(y|H_1)$$

Now what do these signify, in english, or expanded probability notation?

My best bet is that $P(y|H_0) = P(y = 0|H_0) + P(y=1|H_0)$, considering that $Y$ is binary, can take only 0 and 1.

From this the book then arrives at the optimal decision rule which contains this expression.

Answer 1

As requested in comments:

Presumably $P_{Y\mid H}(y\mid H_0)$ represents the probability that $Y=y$ given that the event $H_0$ occurs (i.e. that the null hypothesis $H_0$ is true)