For a finite set \(I\) and \(i\in I\), the coordinate projection \(\mathbb R^I\to \mathbb R\), \(x\mapsto x_i\), has operator norm at most one for the maximum norm.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). The measure \(\mu _{c,v}\) has Lebesgue density \(z\mapsto \prod _i p_v(c_i,z_i)\), where \(p_v(c_i,z_i)\) is the scalar Gaussian density.

Let \(I\) be a finite set, \(c\in \mathbb R^I\), \(v\gt 0\), and \(\mu _{c,v}=\bigotimes _{i\in I}N(c_i,v)\). For a real normed vector space \(E\) and any \(f:\mathbb R^I\to E\), \(\int f\, d\mu _{c,v}=\int _{\mathbb R^I}(\prod _i p_v(c_i,z_i))f(z)\, dz\), where \(p_v\) is the scalar Gaussian density and nonintegrable integrals on either side are assigned zero.

Let \(I\) be finite, \(c\in \mathbb R^I\), and \(s\in \mathbb R\). The map \(y\mapsto (c_i+sy_i)_i\) pushes \(\bigotimes _iN(0,1)\) to \(\bigotimes _iN(c_i,s^2)\), including negative or zero \(s\).

Let \(I\) be finite, \(c\in \mathbb R^I\), \(s\ne 0\), and \(f:\mathbb R^I\to \mathbb R\) measurable. For a standard Gaussian vector \(Y\), \(\mathbb E f(c+sY)=\int _{\mathbb R^I}(\prod _i p_{s^2}(c_i,z_i))f(z)\, dz\), where \(p_{s^2}\) is the scalar Gaussian density and nonintegrable integrals are assigned zero.

For a finite set \(I\), \(c\in \mathbb R^I\), \(v\ge 0\), and \(i\in I\), the function \(z\mapsto (z_i-c_i)^2\) is integrable under \(\bigotimes _jN(c_j,v)\).

For a finite set \(I\), \(c\in \mathbb R^I\), \(v\ge 0\), and \(i\in I\), \(\int (z_i-c_i)^2\, d(\bigotimes _jN(c_j,v))(z)=v\).