Let \(A,D,W\) be real functions on a probability space \((\Omega ,\mu )\) such that \((A,D)\) and \(W\) are measurable up to null sets and \(W\) is independent of \((A,D)\). If \(\psi \colon \mathbb R\to \mathbb R\) is continuous and \(\psi (A+DW)\) is integrable, then \(q(a,d)=\mathbb E_\mu [\psi (a+dW)]\) is integrable under the law \(\lambda \) of \((A,D)\) and \(\mathbb E_\mu [\psi (A+DW)]=\int q(a,d)\, d\lambda (a,d)\).

Let \(A,D,U,V\) be real random variables on a probability space, with \(A,D\in L^4\). Suppose each of \(U,V\) is independent of the pair \((A,D)\) and is measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\), \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, and both \(\psi (A+DU)\) and \(\psi (A+DV)\) are integrable, then \(|\mathbb E\psi (A+DU)-\mathbb E\psi (A+DV)|\le (3C/4)\mathbb E[D^4]\).

Let \(F(x)=\sum _{S\subseteq \{ 1,\ldots ,n\} }a_S\prod _{j\in S}x_j\) be a real multilinear polynomial and fix \(i\in \{ 1,\ldots ,n\} \). Suppose \(F\) has degree at most \(k+1\), and \(X,Y\) are real input families on one probability space, each independent within itself, agreeing as functions at every coordinate other than \(i\). Every input is measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). If \(\psi \in C^4(\mathbb R)\) and \(C\ge 0\) bounds \(|\psi ^{(4)}|\) everywhere, then \(|\mathbb E\psi (F(X))-\mathbb E\psi (F(Y))|\le (C/12)9^{k+1} I_i(F)^2\), where \(I_i(F)=\sum _{S\ni i}a_S^2\).