An extension of the mean value theorem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908573157556224 |
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| author | Lasserre, Jean B |
| author_facet | Lasserre, Jean B |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An extension of the mean value theorem Lasserre, Jean B Optimization and Control 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. |
| title | An extension of the mean value theorem |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.01726 |