Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) be a Boolean function of degree at most \(0\); that is, every set \(S\) with nonzero Fourier–Walsh coefficient \(\hat f(S)\ne 0\) satisfies \(|S|\le 0\). Then \(f\) is constant: for every point \(x\in \{ 0,1\} ^n\) one has \(f(x)=f(x_0)\), where \(x_0\) is the all-\(\mathtt{false}\) point.

Let \(f:\{ 0,1\} ^n\to \mathbb {R}\) be a Boolean function of degree at most \(0\), meaning that every set \(S\) with a nonzero Fourier coefficient \(\hat f(S)\ne 0\) satisfies \(|S|\le 0\). Then, with expectations taken under the uniform measure on \(\{ 0,1\} ^n\),

Fix a natural number \(m\), and let \(a, b, B, C\) be nonnegative real numbers and \(A\) a real number satisfying \(A \le 9^{m+1} a^2\), \(B \le 9^m b^2\), and \(C^2 \le A B\). Then

Let \(f : \{ 0,1\} ^n \to \mathbb {R}\) be a Boolean function of degree at most \(k\), meaning that every nonzero Fourier coefficient \(\hat f(S)\) is supported on a set \(S\) with \(|S| \le k\). Writing \(\mathbb {E}\) for the expectation under the uniform measure on \(\{ 0,1\} ^n\), we then have

Let \(f : \{ 0,1\} ^n \to \mathbb {R}\) be a Boolean function of arity \(n\), let \(p\) be a natural number, and let \(P\) be a probability measure on the Boolean hypercube \(\{ 0,1\} ^n\) that assigns to each singleton \(\{ x\} \) the uniform weight \(2^{-n}\). Then the \(p\)-th moment of \(f\) under \(P\) equals the expectation of \(f^p\) under the uniform measure:

Let \(n\) be a natural number and let \(x\) be a point of the Boolean hypercube \(\{ 0,1\} ^n\). Under the canonical uniform probability measure on \(\{ 0,1\} ^n\), the measure of the singleton \(\{ x\} \), taken as a real number, equals the uniform weight \(2^{-n}\).

Let \(f : \{ 0,1\} ^n \to \mathbb {R}\) be a Boolean function whose Fourier expansion has degree at most \(k\); that is, every nonzero Fourier coefficient \(\hat f(S)\) is supported on a set \(S\) with \(\left\lvert S\right\rvert \le k\). When \(\{ 0,1\} ^n\) is equipped with the uniform probability measure, \(f\) is \(9^k\)-reasonable: its fourth moment is controlled by the square of its second moment via