Lemma 18.34.1666CB6C16A6B7

Proof. Orthonormality gives $\int _ X|C_ v(x,y)|^2\, \mathrm{d}\nu (y)=B_ v(x)$, so the first marginal of $J_ v$ is $\mu _ v$. By symmetry so is the second, and the total mass identity shows that $J_ v$ has mass one. Off $Z_ m$, dividing $|C_ v(x,\cdot )|^2\, \mathrm{d}\nu $ by $B_ v(x)$ gives the squared Riesz-vector density in (18.34.10). Thus the probabilities $\sigma _{v,x}$ are the conditional probabilities of $J_ v$ given the first coordinate, and $\mathcal Q_ v$ is the associated Markov operator. Since these conditional probabilities have mass one and the marginal of $J_ v$ is $\mu _ v$, the operator $\mathcal Q_ v$ fixes constants and preserves integrals; Jensen's inequality shows that it is a contraction of $L^2(\mu _ v)$, and the symmetry of $J_ v$ shows that it is selfadjoint. For $f\in L^2(\mu _ v)$, expanding the finite sum gives

Each of these integrals is independent of the phases, and it exists because $|F_ i\overline{F_ j}|\leq B_ v$ and $f\in L^1(\mu _ v)$. This proves positivity. $\square$