For finite subsets \(S,J\) of a set and a real function \(g\) on finite subsets, if \(S\nsubseteq J\) then \(\sum _{T\subseteq S}(-1)^{|S|-|T|}g(T\cap J)=0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J,S\subseteq [n]\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). Then \(E_J(f_S)(x)=f_S(x)\) if \(S\subseteq J\), and \(E_J(f_S)(x)=0\) otherwise.

For a finite set \(U\) and a real function \(g\) on finite subsets, \(\sum _{S\subseteq U}\sum _{T\subseteq S}(-1)^{|S|-|T|}g(T)=g(U)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S,T\subseteq [n]\), \(f:\Omega \to \mathbb R\), and \(x\in \Omega \). The \(S\)-component of \(f_T\) equals \(f_S(x)\) when \(S=T\), and equals zero otherwise.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(S\subseteq [n]\), \(\rho \in \mathbb R\), and \(f:\Omega \to \mathbb R\). The \(S\)-component of \(T_\rho f\) is \(\rho ^{|S|}f_S\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \mathbb R\). Its orthogonal components satisfy \(\sum _{S\subseteq [n]}f_S(x)=f(x)\) for every \(x\in \Omega \).

Let \(s\) be finite, let \(w_i\ge 0\) for \(i\in s\) satisfy \(\sum _{i\in s}w_i=1\), and let \(F_i\in \mathbb R\) and \(q\ge 1\). Then \(|\sum _{i\in s}w_iF_i|^q\le \sum _{i\in s}w_i|F_i|^q\).

Let \(s\) be finite with weights \(w_i\ge 0\) summing to \(1\), values \(F_i\in \mathbb R\), \(q\gt 0\), and \(0\le c\le 1\). If \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\), then with \(a=\sum _{i\in s}w_iF_i\), \(\sum _{i\in s}w_i|a-c(F_i-a)|^q\le \sum _{i\in s}w_i|F_i|^q\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J\subseteq [n]\), \(f:\Omega \to \mathbb R\), \(q\gt 0\), and \(0\le c\le 1\). If \(|1-cz|^q\le |1+z|^q-q(1+c)z\) for every real \(z\), then \(\mathbb E_P|E_Jf-c(f-E_Jf)|^q\le \mathbb E_P|f|^q\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J\subseteq [n]\), \(f:\Omega \to \mathbb R\), and \(q\ge 1\). Then \(\mathbb E_P|E_Jf|^q\le \mathbb E_P|f|^q\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(k\in \mathbb N\), \(f:\Omega \to \mathbb R\) of degree at most \(k\), and \(c\in \mathbb R\). Then \(x\mapsto f(x)-c\) has degree at most \(k\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\) of degree at most \(k\in \mathbb N\), and \(\operatorname {Var}_P(f)\gt 0\). Then \(k\gt 0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \mathbb R\) with orthogonal components \(f_S\). Then \(\operatorname {Var}_P(f)=\sum _{\varnothing \ne S\subseteq [n]}\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J,S\subseteq [n]\), and \(f:\Omega \to \mathbb R\). The \(S\)-component of \(E_Jf\) equals \(f_S\) when \(S\subseteq J\), and zero otherwise.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), a coordinate \(i\), and \(S\subseteq [n]\). The \(S\)-component of \(L_if\) is \(f_S\) if \(i\in S\), and zero otherwise.

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and a coordinate \(i\). Then \(I_i(f)=\sum _{S\ni i}\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \mathbb R\). Then \(I(f)=\sum _{S\subseteq [n]}|S|\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) and \(f:\Omega \to \{ -1,1\} \). Then \(\operatorname {Var}_P(f)\le 1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(J\subseteq [n]\), and \(f:\Omega \to \mathbb R\). Then \(\mathbb E_P(f-E_Jf)^2=\sum _{S\nsubseteq J}\mathbb E_Pf_S^2\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \mathbb R\), and \(k\in \mathbb N\). Then \((k+1)\sum _{|S|\gt k}\mathbb E_Pf_S^2\le I(f)\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(f:\Omega \to \{ -1,1\} \), \(J\subseteq [n]\), \(k\in \mathbb N\), and \(a,b\in \mathbb R\). If \(\sum _{|S|\gt k}\mathbb E_Pf_S^2\le a\) and \(\sum _{|S|\le k,\ S\nsubseteq J}\mathbb E_Pf_S^2\le b\), then there exists \(g:\Omega \to \{ -1,1\} \) depending only on \(J\) with \(\Pr _P(f\ne g)\le a+b\).