For nonnegative functions on a finite Boolean lattice, the condition for all pairs implies . Coordinate elimination and the two-point four-functions inequality prove it by induction. Indicators give inequalities for the union and intersection of set families. The Boolean-lattice statement appears in Theorem 3.1 of Nicholas Ruozzi's paper.
Suppose nonnegative numbers satisfy , , and . Set , , , . Then and . For , gives ; for , . Adding the two diagonal bounds proves . This lets the four functions theorem sum out one Boolean coordinate while preserving its pointwise hypothesis.
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The Ahlswede–Daykin inequality is a result in information theory that relates to the concept of entropy and the joint distribution of random variables. It provides a connection between the joint entropy of a set of variables and the individual entropies of those variables, specifically in the context of entropy in multiple dimensions. To give a brief overview, let \( X \) and \( Y \) be two discrete random variables with joint distribution.