On page 74 of Lehmann's Testing Statistical Hypothesis, the author writes
Let $P_0$ and $P_1$ be probability distributions possessing densities $p_0$ and $p_1$ respectively w.r.t. a measure $\mu$
...
Let $\alpha\left(c\right) = P_0 \left\{ p_1\left(X\right) > c p_0\left(X\right)\right\}$. Since...
Shouldn't it be $P_0\left\{ p_1 \left(x\right) > c p_0\left(x\right)\right\}$? Since $P_0$ is a measure defined on the Borel $\sigma$-algebra, it makes sense to write $P_0 \left( \left\{x \in \mathbb{R} : p_1 \left(x\right) > c p_0\left(x\right)\right\}\right)$, but it does not make sense to write $P_0\left( \left\{ \omega \in \Omega: p_1 \left(X\left(\omega\right)\right) > c p_0 \left(X \left(\omega\right)\right)\right\}\right)$
Similarly, on page 69,
...called the level of significance, and imposes the condition that
$$P_\theta \left\{ X \in S_1 \right\} \le \alpha$$
Again, $S_1$ is a subset of $\mathbb{R}$ (or $\mathbb{R}^n$). So shouldn't it just be $\mathbb{P}_\theta \left\{S_1\right\} \le \alpha$?
On page 36, the author clearly states that
...Such a measurable real-valued $X$ is called a random variable, and the probability measure it generates over$\left(\mathcal{X},\mathcal{A}\right)$ will be denoted by $P^X$ and called the probability distribution of $X$.
