Let \(A\subseteq \mathbb R\) be measurable, \(c,s\in \mathbb R\), and \(s\ne 0\). For \(Z\sim N(0,1)\) and the Gaussian density \(p_{s^2}(c,z)\) with mean \(c\) and variance \(s^2\), \(\mathbb E1_A(c+sZ)=\int _{\mathbb R}p_{s^2}(c,z)1_A(z)\, dz\).

For \(v\ge 0\), put \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\) when \(v\gt 0\), and \(p_0(c,z)=0\); take division by zero to be zero. For all real \(c,z\), the derivative at \(c\) of \(m\mapsto p_v(m,z)\) is \(((z-c)/v)p_v(c,z)\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For every \(M\ge 0\), there exists a Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(|((z-m)/v)p_v(m,z)|\le b(z)\) for all \(z\in \mathbb R\) and all \(m\) with \(|m|\le M\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). Let \(f:\mathbb R\to \mathbb R\) be measurable with \(|f(z)|\le 1\) for all \(z\). For every \(c\in \mathbb R\), the derivative at \(c\) of \(m\mapsto \int p_v(m,z)f(z)\, dz\) is \(\int ((z-c)/v)p_v(c,z)f(z)\, dz\).

For \(v\ge 0\), put \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\) when \(v\gt 0\), and \(p_0(c,z)=0\); take division by zero to be zero. For real \(c,z\), the derivative at \(c\) of \(m\mapsto ((z-m)/v)p_v(m,z)\) is \(((z-c)^2/v^2-1/v)p_v(c,z)\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For every \(M\ge 0\), there is a Lebesgue-integrable \(b:\mathbb R\to \mathbb R\) such that \(|((z-m)^2/v^2-1/v)p_v(m,z)|\le b(z)\) for all real \(z\) and all \(m\) with \(|m|\le M\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). If \(f:\mathbb R\to \mathbb R\) is measurable and \(|f|\le 1\), then for each real \(c\) the derivative at \(c\) of \(m\mapsto \int ((z-m)/v)p_v(m,z)f(z)\, dz\) is \(\int ((z-c)^2/v^2-1/v)p_v(c,z)f(z)\, dz\).

For every \(c\in \mathbb R\) and \(v\ge 0\), a random variable \(Z\sim N(c,v)\) satisfies \(\mathbb E(Z-c)^2=v\), including the point-mass law when \(v=0\).

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). Let \(c\in \mathbb R\) and \(g,b:\mathbb R\to \mathbb R\). If \(b\) is integrable for \(N(c,v)\) and \(|g(z)|\le b(z)\) for every \(z\), then \(|\int p_v(c,z)g(z)\, dz|\le \int b(z)\, dN(c,v)(z)\), with a nonintegrable integral on the left assigned value zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For any function \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\) and any real \(c\), \(|\int ((z-c)^2/v^2-1/v)p_v(c,z)f(z)\, dz|\le 2/v\), where nonintegrable integrals are assigned zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For any function \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\) and any real \(c\), \(|\int ((z-c)/v)p_v(c,z)f(z)\, dz|\le 1+1/v\), where nonintegrable integrals are assigned zero.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int ((z-c)/v)p_v(c,z)f(z)\, dz\) is globally \(2/v\)-Lipschitz.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int p_v(c,z)f(z)\, dz\) is globally \((1+1/v)\)-Lipschitz.

Let \(v\gt 0\) and \(p_v(c,z)=(2\pi v)^{-1/2}\exp (-(z-c)^2/(2v))\). For measurable \(f:\mathbb R\to \mathbb R\) with \(|f|\le 1\), the function \(c\mapsto \int p_v(c,z)f(z)\, dz\) is continuously differentiable on \(\mathbb R\).