An extension of the mean value theorem
Résumé
Let (Ω, µ) be a measure space with Ω ⊂ 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 dµ = µ(Ω)f (x 0 ) for some x 0 ∈ 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.
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