Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. For every \(\varepsilon \gt 0\) there exists \(C\gt 0\) such that \(\overline I(p)\le \varepsilon +C|p-b|\) for all \(p\in [0,1]\) and \(b\in \{ 0,1\} \).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For a measurable \(A\subseteq \mathbb R^n\), let \(u_\theta (x)=\mathbb E_Y1_A(\cos \theta \, x+\sin \theta \, Y)\) with \(Y\sim \gamma _n\), and put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\) for independent \(X,Y\sim \gamma _n\). For every \(\varepsilon \gt 0\) there exists \(C\gt 0\) such that \(\int \overline I(u_\theta (x))\, d\gamma _n(x)\le \varepsilon +CR_A(\theta )\) for every real \(\theta \).

Let \(\Phi ,\varphi \) be the standard Gaussian distribution function and density, and let \(\overline I(p)=\varphi (\Phi ^{-1}(p))\) for \(0\lt p\lt 1\), with \(\overline I(p)=0\) elsewhere. Let \(\gamma _n\) be standard Gaussian measure on \(\mathbb R^n\). For a measurable \(A\subseteq \mathbb R^n\), let \(u_\theta (x)=\mathbb E_Y1_A(\cos \theta \, x+\sin \theta \, Y)\) with \(Y\sim \gamma _n\), and put \(R_A(\theta )=\Pr (1_A(X)\ne 1_A(\cos \theta \, X+\sin \theta \, Y))\) for independent \(X,Y\sim \gamma _n\). Then \(\int \overline I(u_\theta (x))\, d\gamma _n(x)\to 0\) as \(\theta \to 0\) through real values.