An extension of the mean value theorem

Abstract

Let ($\Omega$, $\mu$) be a measure space with $\Omega$ $\subset$ R d and $\mu$ a finite measure on $\Omega$. We provide an extension of the Mean Value Theorem (MVT) in the form It is valid for non compact sets $\Omega$ and f is only required to be integrable with respect to $\mu$. It also contains as a special case the MVT in the form f d$\mu$ = $\mu$($\Omega$)f (x 0 ) for some x 0 $\in$ $\Omega$, valid for compact connected set $\Omega$ 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.