An extension of the mean value theorem

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.