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According to Wasserman's All of Statistics pg. 92:

A $1 - \alpha$ confidence interval for a paramater $\theta$ is an interval $C_n = (a,b)$ where $a = a(X_1, \ldots , X_n)$ and $b = b(X_1 , \ldots, X_n)$ are functions of the data such that

$$ \mathbb{P}_\theta ( \theta \in C_n ) > 1 - \alpha \text{, for all } \theta \in \Theta $$

...If $\theta$ is a vector then we use a confidence set (such as a sphere or an ellipse) instead of an interval.

Question:

While I understand conceptually what a confidence interval is (i.e., a 95% CI means that 95% of experiments will trap the paramater in the interval), I don't understand how this formality is capturing this concept.

In particular, I don't understand what is meant by the notion of $\mathbb{P}_\theta( \theta \in C_n)$. What is the sample space which $\mathbb{P}$ is drawing from? What is the set $\theta \in C_n$? It seems here $\theta$ is being treated both as a fixed value (from the notation $\mathbb{P}_\theta$) and as a random variable (by the notation $\theta \in C_n$)

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$C_n$ is an interval with random endpoints, denoted $a$ and $b$. Both the endpoints are functions of your sample $X_1, X_2,\ldots, X_n$, and the joint distribution of the $X$'s is parametrized by $\theta$, hence the subscript on $P_\theta$. The parameter $\theta$ that governs this joint distribution is nonrandom, and generally unknown (and the mission of the CI is to capture this unknown parameter). The set $\{\theta\in C_n\}$ is shorthand for $$ \{\omega: a(X_1(\omega), X_2(\omega), \ldots, X_n(\omega)) < \theta < b(X_1(\omega), X_2(\omega), \ldots, X_n(\omega)) \}\tag1 $$ Viewed this way, the event (1) is more a statement about the random endpoints $a$ and $b$, than about the parameter $\theta$: it's asking whether the random left endpoint is less than the number $\theta$ and the random right endpoint is greater than the number $\theta$. In the frequentist treatment of confidence intervals, $\theta$ is not a random variable; in other treatments it is possible to regard $\theta$ as the observed value of a random variable, but that's not what this definition appears to be about.

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