From Wikipedia:
In statistics, the size $\alpha$ of a test is the probability of falsely rejecting the null hypothesis/making a type $1$ error. For a simple hypothesis, $$\alpha=\mathbb{P}(\text{test rejects }H_0|H_0).$$
I am a bit confused because of the last equation. I thought that $H_0$ should be imagined to be a proposition that is either true or false independently of the outcome of the experiment. In particular, $H_0$ is not an event, is it? So the notation (which suggests that $\alpha$ is a conditional probability) is misleading, isn't it? I assume that the author meant that $\alpha=\mathbb{P}(\text{test rejects }H_0)$ if $H_0$ is true and otherwise the size is simply not defined (or equal to zero, as no type 1 error can occur) - is that correct?
This is not a rant, I just want to make sure that I understand the definition correctly.
Suppose one has a family of probability measures $\mathcal P$. Let $\mathcal P_0 \sqcup \mathcal P_1$ be a partition of $\mathcal P$. A test $T$ is a statistic taking values in $\{0,1\}$. One way to evaluate the performance of a test is its size $\alpha$: $$\alpha \overset{\text{def}}{=} \sup_{P\in \mathcal P_0} P(T=1)$$ It is usually intractable to compute $\alpha$, and when people talk about the size of a test, they really mean the limiting size of the test. Namely, $T_n = T_n(X_1,\dots,X_n)$ and $$\alpha \overset{\text{def}}{=}\lim_{n\to\infty} \sup_{P\in \mathcal P_0}P(T_n = 1)$$