Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(u_\theta (x)=\gamma _n\{ y:\cos \theta \, x+\sin \theta \, y\in A\} \) for every real \(\theta \) and every \(x\).

Let \(\Omega ,\Xi \) be measurable spaces, \(A\subseteq \Xi \) a measurable set, and \(f,g:\Omega \to \Xi \) measurable maps. Then \(\{ z:\neg (f(z)\in A\ \Longleftrightarrow \ g(z)\in A)\} \) is measurable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For independent \(X,Y\sim \gamma _n\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(A\) is measurable, then \(\int |u_\theta (x)-1_A(x)|\, d\gamma _n(x)=R_A(\theta )\) for every real \(\theta \).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(0\le u_\theta (x)\le 1\) for every real \(\theta \) and every \(x\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). If \(A\) is measurable, then \(\int |u_\theta -1_A|\, d\gamma _n\to 0\) as \(\theta \to 0\) through real values.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), the function \(u_\theta \) is measurable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), the function \(u_\theta \) is \(\gamma _n\)-integrable.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For measurable \(A\) and every real \(\theta \), \(\int u_\theta \, d\gamma _n=\gamma _n(A)\).

Let \(E\) be a measurable space and \(\mu \) an s-finite measure on \(E\). The map \(((a,b),(c,d))\mapsto ((a,c),(b,d))\) preserves the measure \((\mu \otimes \mu )\otimes (\mu \otimes \mu )\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For any real \(u\), if \(X_1,X_2,Y_1,Y_2\) are independent with law \(\gamma _n\), then \((\cos u\, X_1+\sin u\, Y_1,\cos u\, X_2+\sin u\, Y_2)\) has law \(\gamma _n\otimes \gamma _n\).

Let \(\mu \) be a finite measure on a measurable space \(E\), and let \(C,D\subseteq E\) be arbitrary sets. Then \(|\mu (C)-\mu (D)|\le \mu (C\mathbin {\triangle } D)\), where all values are viewed as finite real numbers.

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). Let \(A\) be measurable, \(u,v\in \mathbb R\), and \(x,x\prime \in \mathbb R^n\). For independent \(Y,Y\prime \sim \gamma _n\), \(|u_u(x)-u_u(\cos v\, x+\sin v\, x\prime )|\le \Pr (1_A(\cos u\, x+\sin u\, Y)\ne 1_A(\cos u(\cos v\, x+\sin v\, x\prime )+\sin u(\cos v\, Y+\sin v\, Y\prime )))\).

Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For \(A\subseteq \mathbb R^n\) and \(\theta \in \mathbb R\), put \(u_\theta (x)=\int 1_A(\cos \theta \, x+\sin \theta \, y)\, d\gamma _n(y)\). For independent \(X,Y\sim \gamma _n\), put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\). If \(A\) is measurable, then for every \(u,v\in \mathbb R\), \(\mathbb E_{X,Y}|u_u(X)-u_u(\cos v\, X+\sin v\, Y)|\le R_A(v)\).