For a finite set \(\Omega \) and commutative unital ring \(R\), every \(R\)-linear functional on the function algebra has a unique weight function \(p:\Omega \to R\), given by \[\omega (f)=\sum _{x\in \Omega }p(x)f(x),\qquad p(x)=\omega (\delta _x).\] Conversely, every such weight function defines an \(R\)-linear functional. It is normalized exactly when \(\sum _xp(x)=1_R\).

When \(\Omega =\varnothing \), the sole functional is the zero map from the zero algebra and its total weight is \(0_R\); it is normalized exactly when \(R\) is the zero ring. When \(\Omega \ne \varnothing \), a chosen evaluation is normalized over every \(R\). Uniform weights \(p(x)=c\) on a set of size \(n>0\) are normalized exactly when \((n1_R)c=1_R\). Such normalized uniform weights exist exactly when \(n1_R\) is a unit, and then the unique coefficient is \(c=(n1_R)^{-1}\). Thus failure of uniform normalization does not imply failure of all normalized functionals.

More generally, for raw weights \(w(x)\) with total \(Z=\sum _xw(x)\), a common rescaling \(p(x)=cw(x)\) is normalized exactly when \(cZ=1_R\). This is possible exactly when \(Z\) is a unit, and then \(c=Z^{-1}\) is unique. No division by a nonunit is defined by this formula. In the zero ring the sole element is a unit, so the same criterion applies. For real nonnegative weights, \(Z>0\) exactly when at least one weight is positive; all-zero weights cannot be normalized by rescaling.

This is the indicator-basis argument of finite function duality, with the unital algebra convention of [stacks2017stacks, Section 22.2], Tag 09JF. For real nonnegative normalized weights it gives the finite expectation. No positivity or division is needed for the ring-valued classification.