Let \(I,J,B\) be finite sets, let \(B\) carry a measurable structure with measurable singletons, and let \(\mu _{ij}\) be sigma-finite measures on \(B\). The bijection \(B^{I\times J}\to (B^J)^I\) sending \(x\) to the array \(i\mapsto (j\mapsto x(i,j))\) pushes \(\bigotimes _{(i,j)}\mu _{ij}\) to \(\bigotimes _i\bigotimes _j\mu _{ij}\).

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 \(I\) be finite and \(f:\{ 0,1\} ^I\to [0,1]\). Under the independent uniform bit law, \(\overline I(\mathbb E f)\le \mathbb E\sqrt{\overline I(f)^2+\sum _{i\in I}(D_i f)^2}\), where \(D_i f(x)=(f(x^{i\leftarrow 0})-f(x^{i\leftarrow 1}))/2\).

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 \(I,J\) be finite and \(f:\{ 0,1\} ^{I\times J}\to [0,1]\). Under independent uniform bits, \(\overline I(\mathbb E f)\le \mathbb E\sqrt{\overline I(f)^2+\sum _{i\in I}\sum _{j\in J}(D_{ij}f)^2}\), where \(D_{ij}f(x)=(f(x^{ij\leftarrow 0})-f(x^{ij\leftarrow 1}))/2\).