A mean-value theorem for positive linear functionals (Q2313395)

Let \(\mathcal F\) be a linear space of functions, defined on some segment \([a,b]\), that is closed under multiplication. A linear functional \(A\) on \(\mathcal F\) is called strictly positive if \begin{itemize}\item[(1)] \(A(h)\ge0\) whenever \(h\in\mathcal F\) is \(\ge0\) almost everywhere (a.e.), and if \item[(2)] \(h\ge0\) a.e.\ and \(A(h)=0\) imply \(h=0\) a.e. \end{itemize} The authors prove a mean-value theorem concerning a strictly positive \(A\) (Theorem 1): If \(f\in\mathcal F\) has the Darboux property in \((a,b)\) and \(g\in\mathcal F\) is almost everywhere nonnegative, then there exists a \(c\in(a,b)\) such that \(A(fg)=f(c)\,A(g)\). As corollaries, a couple of generalised mean-value theorems for integrals are obtained.