[Math] Do you know important theorems that remain unknown

big-listreference-requestsoft-question

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost
nobody knows about them. If you provide an answer, please:

  1. State only one theorem per answer. When people will vote on your answer they will vote on a particular theorem.

  2. Provide a careful statement and all necessary definitions so that a well educated graduate student
    working in a related area would understand it.

  3. Provide references to the original paper.

  4. Provide references to more recent and related work.

  5. Just make your answer useful so other people in the mathematical community can use it right away.

  6. Add comments: how you discovered it, why it is important etc.

  7. Please, make sure that your answer is written at least as carefully as mine. I did invest quite a lot of time writing my answers.

As an example I will provide three answers to this question. I discovered these results while searching for papers related to the questions I was working on.

Best Answer

Monotony is a superfluous hypothesis in the Monotone convergence theorem for Lebesgue integral. In fact the following is true.

Theorem - Let $(X, \tau, \mu)$ be a measurable space, $f_n : X \rightarrow [0,\infty]$ a sequence of measurable functions converging almost everywhere to a function $f$ so that $f_n \leq f$ for all $n$. Then $$\lim_{n\rightarrow \infty} \int_X f_n d\mu = \int_X f d\mu.$$

Proof: $$ \int_X f d\mu = \int_X \underline{\lim} \, f_n d\mu \leq \underline{\lim} \int_X f_n d\mu \leq \overline{\lim} \int_X f_n d\mu \leq \int_X f d\mu. $$

I learnt this result from an article by J.F. Feinstein in the American Mathematical Monthly, but I never saw it in any textbook. Since the Monotone convergence theorem is important, I wish to argue that this is also an important theorem. Here is an illustration.

Let $(X, \tau, \mu)$ be a measurable space, $f : X \rightarrow [0,\infty]$ a measurable function. Then $$ \int_X f d\mu = \lim_{r \rightarrow 1, r>1} \sum_{n\in {\bf Z}} r^n \mu\Bigl( f^{-1}([r^n, r^{n+1}))\Bigr). $$ Neither the dominated nor the monotone convergence theorem apply here. Note that this is a way to define the Lebesgue integral of nonnegative functions. Computing integrals by geometrically dividing the $x$ axis is due to Fermat.

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