Exercise 12.12: Probability Spaces and Kernels
Take \(\mathbb{H} = L^2(\P)\) and \(\phi \colon A \mapsto \ind_A - \E[\ind_A]\). Then \(k(A, B) = \E [(\ind_A - \E[\ind_A])(\ind_B - \E[\ind_B])] = \P(A \cap B) - \P(A) \P(B)\).
Take \(\mathbb{H} = L^2(\P)\) and \(\phi \colon A \mapsto \ind_A - \E[\ind_A]\). Then \(k(A, B) = \E [(\ind_A - \E[\ind_A])(\ind_B - \E[\ind_B])] = \P(A \cap B) - \P(A) \P(B)\).