Let S(x) be a family of confidence sets for a real-valued parameter , and let [S(x)] denote

Let S(x) be a family of confidence sets for a real-valued parameter θ, and let µ[S(x)] denote its Lebesgue measure. Then for every fixed distribution Q of X (and hence in particular for Q = Pθ0 where θ0 is the true value of θ)
EQ{µ[S(X)]} = 
θ=θ0 Q{θ ∈ S(X)} dθ
provided the necessary measurability conditions hold.
[The identity is known as the Ghosh-Pratt identity; see Ghosh (1961) and Pratt (1961a). To prove it, write the expectation on the left side as a double integral, apply Fubini’s theorem, and note that the integral on the right side is unchanged if the point θ = θ0 is added to the region of integration.]