An extension of the mean value theorem
Jean B Lasserre
TL;DR
The paper extends the classical Mean Value Theorem by relaxing the requirements of compactness and continuity, requiring only a finite measure space $(\Omega,\mu)$ and $f\in L^1(\Omega,\mu)$ with $\mu(\Omega)>0$. It leverages Richter's theorem to obtain a two-point atomic representation of the integral: there exist $x_0,x_1\in\Omega$ and $\lambda\in[0,1]$ such that $\int_\Omega f\,d\mu = \mu(\Omega)\,[\,\lambda f(x_0) + (1-\lambda) f(x_1)\,]$. In the classical compact-continuous case, the expression reduces to $\int_\Omega f\,d\mu = \mu(\Omega)\,f(x_0)$; in general, the average is a convex combination of two point evaluations. The work situates this result within Richter–Tchakaloff theory and notes historical commentary on overlooked aspects of the theorems, suggesting potential applications in moment problems and cubature methods.
Abstract
Let ($Ω$, $μ$) be a measure space with $Ω$ $\subset$ R d and $μ$ a finite measure on $Ω$. We provide an extension of the Mean Value Theorem (MVT) in the form It is valid for non compact sets $Ω$ and f is only required to be integrable with respect to $μ$. It also contains as a special case the MVT in the form f d$μ$ = $μ$($Ω$)f (x 0 ) for some x 0 $\in$ $Ω$, valid for compact connected set $Ω$ and continuous f . It is a direct consequence of Richter's theorem which in turn is a non trivial (overlooked) generalization of Tchakaloff's theorem, and even published earlier.