Exercise 12.12: Probability Spaces and Kernels

chapter 12

Take $\mathbb{H}=L^2(\mathbb{P})$ and $\varphi :A\mapsto {\mathbb{1}}_A-\mathbb{E}[{\mathbb{1}}_A]$. Then $k(A,B)=\mathbb{E}[({\mathbb{1}}_A-\mathbb{E}[{\mathbb{1}}_A])({\mathbb{1}}_B-\mathbb{E}[{\mathbb{1}}_B])]=\mathbb{P}(A\cap B)-\mathbb{P}(A)\mathbb{P}(B)$.