Chebyshev's Sum Inequality/Continuous

Theorem

Let $u, v: \closedint 0 1 \to \R$ be integrable functions.


This article, or a section of it, needs explaining.
In particular: Is it important in what sense integrable: Darboux, Lebesgue, etc.?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Let $u$ and $v$ both be either increasing or decreasing.

Then:

$\ds \paren {\int_0^1 u \rd x} \cdot \paren {\int_0^1 v \rd x} \le \int_0^1 u v\rd x$


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Sources

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.