Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(S\subseteq [n]\), and \(f:\Omega \to \mathbb R\). Then \(E_Sf(x)=\sum _y[\prod _{i\in S}\mathbf1_{x_i=y_i}\prod _{i\notin S}w_i(y_i)]f(y)\) for every \(x\in \Omega \).

For a finite set \(I\), \(S\subseteq I\), and real functions \(a,b\) on \(I\), \(\sum _{T\subseteq S}(-1)^{|S|-|T|}\prod _{i\in T}a_i\prod _{i\notin T}b_i=\prod _{i\in S}(a_i-b_i)\prod _{i\notin S}b_i\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(S\subseteq [n]\), and \(f:\Omega \to \mathbb R\). Its orthogonal component satisfies \(f_S(x)=\sum _y[\prod _{i\in S}(\mathbf1_{x_i=y_i}-w_i(y_i))\prod _{i\notin S}w_i(y_i)]f(y)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with coordinate weights \(w_i\), \(\rho \in \mathbb R\), and \(f:\Omega \to \mathbb R\). Then \(T_\rho f(x)=\sum _y[\prod _i(\rho \mathbf1_{x_i=y_i}+(1-\rho )w_i(y_i))]f(y)\) for every \(x\in \Omega \).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(0\le \rho \le 1\), and \(f:\Omega \to \mathbb R\). Then \(\| T_\rho f\| _{1,P}\le \| f\| _{1,P}\).