A \(1 - \alpha\) confidence interval for a parameter \(\theta\) is an interval \(C_n = (a,b)\) where \(a=a(X_1,\dotsc,X_n)\) and \(b=b(X_1,\dotsc,X_n)\) are functions of the data such that \(P(\theta \in C_n) = 1 - \alpha\).
The action space is \(\mathcal{A}=\Theta\).
\(L(\theta,a) = \left\{ \begin{array}{ll} 0 & \mbox{if } |\theta - a| \leq \delta \\ 1 & \mbox{if } |\theta - a| > \delta \\ \end{array} \right.\)
A confidence interval is not a probability statement about \(\theta\), as \(\theta\) is a fixed quantity, not a random variable.
A \(1 - \alpha\) confidence interval means that \(1 - \alpha\) times when you construct confidence interval it will contain \(\theta\).