For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is continuous, then \(S_\eta \psi \) is differentiable at every real \(x\) with derivative \((\psi (x+\eta )-\psi (x))/\eta \).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). For each integer \(m\ge 0\), if \(\psi \) is \(C^m\) on \(\mathbb R\), then \(S_\eta \psi \) is \(C^{m+1}\) on \(\mathbb R\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(m\ge 0\) is an integer and \(\psi \in C^m(\mathbb R)\), then \((S_\eta \psi )^{(m+1)}(x)=(\psi ^{(m)}(x+\eta )-\psi ^{(m)}(x))/\eta \) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz for \(c\ge 0\), then \(|(S_\eta \psi )\prime (x)|\le c\) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \colon \mathbb R\to \mathbb R\) is \(c\)-Lipschitz with \(c\ge 0\), then \(S_\eta \psi \) is also \(c\)-Lipschitz.

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz for \(c\ge 0\), then \(|S_\eta \psi (x)-\psi (x)|\le c\eta \) for every real \(x\).

For \(\eta \gt 0\) and a real function \(\psi \), write \(S_\eta \psi (x)=\eta ^{-1}\int _0^\eta \psi (x+t)\, dt\). If \(\psi \) is \(c\)-Lipschitz with \(c\ge 0\) and \(m\ge 0\) is an integer, put \(\phi =S_\eta ^{m+1}\psi \), where the exponent denotes repeated application of \(S_\eta \). Then \(\phi \) is \(C^{m+1}\) and \(c\)-Lipschitz, and for every real \(x\), \(|\phi (x)-\psi (x)|\le (m+1)c\eta \) and \(|\phi ^{(m+1)}(x)|\le c(2/\eta )^m\).

For every \(\eta \gt 0\), \(c\ge 0\), and \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), there exists \(\phi \in C^4(\mathbb R)\) such that \(|\phi (x)-\psi (x)|\le 4c\eta \) and \(|\phi ^{(4)}(x)|\le 8c/\eta ^3\) for all real \(x\).

Let \(f,g\colon \Omega \to \mathbb R\) be functions on a probability space \((\Omega ,\mu )\), with \(f\) integrable and \(g\) measurable up to a null set. If a real constant \(D\) satisfies \(|f(\omega )-g(\omega )|\le D\) for every \(\omega \), then \(|\int f\, d\mu -\int g\, d\mu |\le D\).

Let \(U\in L^4(\mu )\) and \(V\in L^4(\nu )\) be real inputs on probability spaces, and let \(B\ge 0\). Suppose that for every \(\phi \in C^4(\mathbb R)\) and \(C\ge 0\) bounding \(|\phi ^{(4)}|\) everywhere, \(|\mathbb E_\mu \phi (U)-\mathbb E_\nu \phi (V)|\le CB\). For every \(c\ge 0\), \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), and \(\eta \gt 0\), \(|\mathbb E_\mu \psi (U)-\mathbb E_\nu \psi (V)|\le 8c\eta +(8c/\eta ^3)B\).

Let \(U\in L^4(\mu )\) and \(V\in L^4(\nu )\) be real inputs on probability spaces, and let \(B\ge 0\). Suppose that for every \(\phi \in C^4(\mathbb R)\) and \(C\ge 0\) bounding \(|\phi ^{(4)}|\) everywhere, \(|\mathbb E_\mu \phi (U)-\mathbb E_\nu \phi (V)|\le CB\). For every \(c\ge 0\) and \(c\)-Lipschitz function \(\psi \colon \mathbb R\to \mathbb R\), \(|\mathbb E_\mu \psi (U)-\mathbb E_\nu \psi (V)|\le 16cB^{1/4}\), including \(B=0\).