Do you know important theorems that remain unknown? 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:


*

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

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

*Provide references to the original paper.

*Provide references to more recent and related work.

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

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

*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.
 A: The following result of Hörmander [2] (see also Theorem 2.5.6 in [1]), plays a significant role in harmonic analysis since all convolution type operators
are translation invariant.
Definition. We say that a bounded linear operator $T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is translation invariant if
$T(\tau_y f)=\tau_y(Tf)$ for all $f\in L^p(\mathbb{R}^n)$ and all $y\in\mathbb{R}^n$,
where $(\tau_y f)(x)=f(x+y)$.

Theorem (Hörmander).
If $T:L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$, $1\leq p<\infty$, $1\leq q\leq\infty$ is non-zero and translation invariant, then $q\geq
 p$.

The proof is simple and well known.
The argument does not generalize to the case of $p=\infty$. 
However, the argument still works if we replace $L^\infty$ by $L^\infty_0$ which is the subspace of $L^\infty$ consisting 
of functions that converge to $0$ at infinity. In that case Hörmander proved the following result:

Theorem (Hörmander).
If $T:L^\infty_0(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is non-zero and translation invariant, then $q=\infty$.

Hörmander (see p.97 in [2]), calls this result somewhat incomplete for $p=\infty$.
I was quite curious about the case $p=\infty$ and since I could not find an answer, I discussed it with several mathematicians. As a result of cooperation with 
M. Bownik, F. L. Nazarov and P. Wojtaszczyk we finally proved the following result:

Theorem.
If $T:L^\infty(\mathbb{R}^n)\to L^q(\mathbb{R}^n)$ is non-zero and translation invariant, then $q\geq 2$. On the other hand, there is
  a non-zero translation invariant operator 
  $T_1:L^\infty(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\cap
 L^\infty(\mathbb{R}^n)$. It follows that $T_1:L^2(\mathbb{R}^n)\to
 L^q(\mathbb{R}^n)$ is bounded for all $2\leq q\leq\infty$.

I was very excited about the result and our proof. However, a few days later I got an e-mail from M. Bownik who told me that that the result had already been proved by 
Liu and van Rooij [3]! Despite the fact that this paper solves a problem of Hörmander, it has only one citation according to MathSciNet. 
Many textbooks in harmonic analysis quote the result of Hörmander, but nobody mentions the beautiful result of Lui and van Rooij!
I cannot resist and I have to recall  my e-mail conversation with Nazarov:
Dear Fedja, Bad news. The problem was solved in 1974  by Liu and Van Rooij.....
Dear Piotr, Actually, this is a wonderful news: instead of going through the
painstaking and time consuming proofreading and submission
process, we can just relax and think of something else :-).
Fore more details see slides: https://sites.google.com/view/piotr-hajasz/research/presentations
[1] L. Grafakos,
Classical Fourier analysis.
Second edition. Graduate Texts in Mathematics, 249. Springer, New York, 2008. 
[2] L. Hörmander,
Estimates for translation invariant operators in $L^p$ spaces. 
Acta Math. 104 (1960), 93-140. https://doi.org/10.1007/BF02547187 https://link.springer.com/article/10.1007%2FBF02547187
[3] T. S. Liu, A. C. M. van Rooij,
Translation invariant maps $L^\infty(G)\to L^p(G)$.
Nederl. Akad. Wetensch. Proc. Ser. A 77 = Indag. Math. 36 (1974), 306-316. https://doi.org/10.1016/1385-7258(74)90021-3
A: Acknowledgment: Let me acknowledge Theo Buehler from whom I have learnt about the theorem to be presented, however since Theo seems to be absent from MO, let me post it by myself.

Lemma (Zabreiko, 1969) Let $X$ be a Banach space and let $p\colon X \to [0,\infty)$ be a seminorm. Suppose that for every absolutely convergent series $\sum_{n=1}^\infty x_n$ in $X$ we have
$$
p\left(\sum_{n=1}^\infty x_n\right) \leqslant \sum_{n=1}^\infty p(x_n) \in [0,\infty].
$$
Then $p$ is continuous. That is, there is a constant $C\geqslant 0$ such that $p(x)\leqslant C\Vert x\Vert$ for all $x\in X$.

Now, using Zabreiko's lemma you may easily recover the open mapping theorem, Banach's bounded inverse theorem, the uniform boundedness principle, and closed graph theorem. For more details see this fantastic post.
Original references:

*

*П. П. Забрейко, Об одной теореме для полуаддитивных функционалов, Функциональный анализ и его приложения, 3:1 (1969), 86–88.


*P. P. Zabreiko, A theorem for semiadditive functionals, Functional
analysis and its applications 3 (1), 1969, 70-72).
(MathSciNet review.)
A: Here is one little-known and one completely unknown result.
The little known result is the Mean Motion Theorem.  This says that for all real numbers $\lambda_j$
and all complex numbers $a_j$ the following limit exists:
$$m:=\lim_{t\to+\infty}\phi(t)/t,\quad\mbox{where}\quad \phi(t)=\arg\sum_{j=1}^na_je^{i\lambda_jt},$$
where $t$ is real. 
(There is a natural way to define what happens to the $\arg$ at the zeros, but 
there is not much loss in generality if one assumes for simplicity that the sum has no real zeros).
This result was conjectured by Lagrange, coming from celestial mechanics, and was proved in full generality by the
combined efforts of H. Weyl, P. Hartman and A. Wintner in the 1930s.
The final result, without any restrictions on $\lambda_k,a_k$ is
due to B. Jessen and H. Tornehave in 1945.  It seems that the subject was forgotten after the 1940s.
The completely unknown result is a much stronger statement for $n=3$ under some additional conditions on $\lambda_j$ and $a_j$, namely that
$$\phi(t)=mt+O(1).$$
This is due to Piers Bohl in 1909.
I have never seen any reference on this stronger result, or any discussion of
possible generalization to larger $n$.
Weyl, Wintner and Hartman refer to Bohl proving the $n=3$ case of their results, the first non-trivial case, but do not discuss the $O(1)$. Favorov's paper from 2008 has Bohl's paper in the reference list but also does not discuss the $O(1)$.  In fact I have not seen ANY mention
of a more precise error term than $o(t)$ in the literature. A number of papers GENERALIZE
the mean motion theorem to infinite sums. But nobody addresses the improvement
of the error term. Here is another piece of evidence that the result is "completely unknown": Precise form of the mean motion theorem.
Remark on references. The only book I know which addresses the subject is Sternberg's 1969 book.  (This book has a rare distinction: it is not reviewed in Mathscinet:-)  The whole first chapter of the book explains the historical background: the problem is evidently related to constructing a calendar:-)
Weyl's 1938 paper is very well written, fortunately in English, and accessible to a non-specialist.  If you can read German or Russian, Bohl's paper is also good reading, it is completely elementary.  I suspect that nobody reads Bohl since his result has been "superseded" by Weyl and Co. It does not help that it
was published in German.
References in chronological order:
Bohl, P., Über ein in der Theorie der säkularen Störungen vorkommendes Problem. 
J. für Math. 135, 189-283 (1909).  (There is a Russian translation which is difficult to obtain, so I post it here for the benefit of this community.)
Weyl, Hermann, Mean motion, Amer. J. Math. 60, 889-896 (1938)
B. Jessen and H. Tornehave, Mean motions and zeros of almost periodic functions.
Acta Math. 77, (1945). 137–279. 
S. Sternberg, Celestial mechanics, Part 1, W. A. Benjamin, NY, 1969.
Favorov, S. Yu., Lagrange's mean motion problem, Algebra i Analiz 20 (2008), no. 2, 218--225; translation in St. Petersburg Math. J. 20 (2009), no. 2, 319–324.  MR2424001 
A: In 1942 A. Sard [3] (see also [4]) proved the following theorem.

Theorem (Sard). Let $f:M^m\to N^n$ be of class $C^k$, and let $S={\rm crit}\, f$.  If $k> \max (m-n, 0)$, then
  $\mathcal{H}^n(f(S))=0$.

Here $\mathcal H^s$ stands for the $s$-dimensional
Hausdorff measure (we shall follow the convention that
$\mathcal H^s$ is the counting measure for all $s\leq 0$) and, for a
$C^1$ mapping $f\colon M^m\to N^n$,
${\rm crit}\, f$ denoted the set of critical points of $f$.
It is well known that the assumptions of Sard's theorem
are optimal within the scale of $C^k$ spaces. 
Now, several years after Sard's paper, A.Ya.Dubovitskii [1]
obtained a more general, better result.

Theorem (Dubovitskii). Let $f\colon M^m\to N^n$ be a mapping of class $C^k$. Set $s=m-n-k+1$. Then $$ \mathcal H^s (f^{-1}(y) \cap
{\rm crit}\, f) = 0  \quad \text{for $\mathcal H^n$ a.e. $y\in N^n$.}
$$

If $k> \max (m-n, 0)$, Sard's theorem follows from that of Dubovitskii. 
Dubovitskii, like a large number of mathematicians in the Soviet
Union of that time, was isolated from the West and from the new
results of western mathematics. He does not quote Sard's paper. On
pages 398-402 of [1] he gives a variant of Whitney's
example for the sharpness of the Sard theorem, and an example of a function $f\in C^k\bigl((0,1)^m,(0,1)^n\bigr)$ such that all sets $f^{-1}(y)\cap
{\rm crit}\, f$ have $(m-n-k)$-dimensional Hausdorff measure greater than
zero, where $m,k,n$ are positive integers such that $m-n-k>0$. He
attributes the first example to Menshov but gives no reference,
and acknowledges Menshov, Novikov, Kronrod and Landis in his
Introduction.
The result of Dubovitskii remained unknown until 2005
when a new proof and some generalizations were published in [2]. It is very surprising since his paper was quoted in Milonor's, Topology from the Differentiable Viewpoint
(p. 10).
[1] A. Ya. Dubovickii, On the structure of level sets of
differentiable mappings of an $n$-dimensional cube into a
$k$-dimensional cube. (Russian) Izv. Akad. Nauk SSSR. Ser.
Mat. 21 (1957), 371-408.
[2] B. Bojarski, P. Hajłasz, P., P. Strzelecki, Sard's theorem for mappings in Hölder and Sobolev spaces. Manuscripta Math. 118 (2005), no. 3, 383–397.
[3] A. Sard, The measure of the critical values of
differentiable maps. Bull. Amer. Math. Soc. 48
(1942), 883-890.
[4] S. Sternberg, Lectures On Differential Geometry.
Prentice Hall, 1964.
Personal comment. I proved the result of Dubovitskii as an undergraduate student. When I discovered that it had already been published, I was quite devastated. I had waited 15 years before I decided to publish it. I am happy I did publish it. Not because of a new `modern' proof and some generalizations that I and my collaborators were able to obtain, but because the old result of Dubovitskii has been brought to public and gained a proper recognition. This comment is related to an answer that I gave to another post.
A: My answer is inspired by the one of coudy: how many scientists who deal with the Lebesgue integral on a daily basis know that there exists a necessary and sufficient condition for the passage to the limit under the integral symbol? I learned about it while reading some papers on the history of Italian mathematics written by the late Gaetano Fichera: the result, stated in modern language ([4], Ch. VIII, pp. 110-128), is reported below.
Definition 1. Let $(E,\mathcal{E})$ be a measure space and $\phi:\mathcal{E}\to\overline{\mathbb{R}}$ a numerical set function:
$\phi$ is called exhaustive if
$$
\lim_n\phi(A_n)=0
$$
for all families $\{A_n\}$ of pairwise disjoint sets in $\mathcal{E}$.
Definition 2. Let $(E,\mathcal{E})$ be a measure space and $H$ a set (and thus possibly a family) of numerical set functions defined on $\mathcal{E}$: $H$ is called uniformly exhaustive if the numerical set function
$$
A\mapsto\sup_{\phi\in H} \vert\phi(A)\vert\;\text{ is exhaustive.}
$$
Cafiero's theorem (on the passage to the limit under the integral).
Let $(E,\mathcal{E})$ be a measure space, $(\mu_n)_{n\geq 1}$ be a sequence of real measures and $(f_n)_{n\geq 1}$ be a sequence of real functions such that $f_n\in\mathcal{L}^1(\vert\mu_n\vert)$ for all $n$ (here the notation $\vert\mu\vert$ identifies the variation of the measure $\mu$). Suppose moreover that the following pointwise limits exist
\begin{split}
\lim_{n\to\infty} \mu_n &=\mu\\
\lim_{n\to\infty} f_n &=f
\end{split}
where $\mu$ and $f$ are respectively a real measure and a real function.
Then
$$
\lim_{n\to\infty} \int f_n\mathrm{d}\mu_n = \int f \mathrm{d}\mu\iff\text{$(f_n\cdot\mu_n)_{n\geq 1}$ is uniformly exaustive.}
$$
The result was originally proved by Cafiero in [1] (see also book [2], ch. VII, §2 pp. 377-392), who generalized the concept of uniform additivity introduced before and independently by Renato Caccioppoli and Vladimir Dubrovskii: that theorem includes the ones of Nykodym, Vitali, Hahn and Saks and an earlier result of Gaetano Fichera [3], where a necessary and sufficient condition was proved for the integral with respect to a given fixed measure. The work of Cafiero is cited in the bibliography of the treatise on linear operators by Dunford and Schwartz but, to my knowledge, the only English reference discussing (very briefly) his contribution is the recent treatise of Vladimir Bogachev.
[1] Cafiero, F. (1953), "Sul passaggio al limite sotto il segno d'integrale per successioni d'integrali di Stieltjes-Lebesgue negli spazi astratti, con masse variabili con gli integrandi [On the passage to the limit under the integral symbol for sequences of Stieltjes–Lebesgue integrals in abstract spaces, with masses varying jointly with integrands]" (Italian), Rendiconti del Seminario Matematico della Università di Padova, 22: 223–245, MR0057951, Zbl 0052.05003.
[2] Cafiero, F. (1959), Misura e integrazione [Measure and integration] (Italian), Monografie matematiche del Consiglio Nazionale delle Ricerche 5, Roma: Edizioni Cremonese, pp. VII+451, MR0215954, Zbl 0171.01503.
[3] Fichera, G. (1943), "Intorno al passaggio al limite sotto il segno d'integrale" [On the passage to the limit under the integral symbol] (Italian), Portugaliae Mathematica, 4 (1): 1–20, MR0009192, Zbl 0063.01364.
[4] Letta, G. (2013), Argomenti scelti di Teoria della Misura [Selected topics in Measure Theory], (in Italian) Quaderni dell'Unione Matematica Italiana 54, Bologna: Unione Matematica Italiana, pp. XI+183, ISBN 88-371-1880-5, Zbl 1326.28001.
A: The following seems, to me, to be "the nicest theorem that does not have a special name". It is a wonderful blend of Topology, Geometry and Analysis. Moreover, it has a short and simple statement, involving only the notions of Euler characteristic and of a Lie group.

Theorem: The Euler characteristic of a connected nontrivial Lie group is zero!

The proof is also simple, being an application of Lefschetz fixed point theorem: Since the left translation endomorphism of a Lie group $G$ (by a non-trivial element) has no fixed points, Lefschetz numbers are homotopy invariant, and the Lefschetz number of the identity map is the Euler characteristic of $G$, the result follows (note that compactness of the Lie group is not really an issue).
Of course, the result (and the nice short proof) is known, but I think it should be much better known, and part of a lot of standard books.
A: Ionin–Pestov theorem is not very well known, but it deserves to be included in standard introductory texts on differential geometry of curves.
It gives the simplest meaningful example of a local-to-global theorem which is what differential geometry is about.
Theorem. Assume that a plane region $F$ is bounded by a simple loop with curvature at most $1$. Then $F$ contains a unit disc.

The original reference:

*

*Пестов, Г. Г., Ионин В. К. О наибольшем круге, вложенном в замкнутую кривую // Доклады АН СССР. — 1959. — Т. 127, № 6.

We used it in our textbook What is differential geometry.
A: The Lusin theorem says that a measurable function coincides with a continuous function away from a set of measure less than $\varepsilon$. A result of Federer says that an a.e. differentiable function coincides with a $C^1$ function away from a set of measure less than $\varepsilon$. So such functions have a $C^1$
Lusin property. Imomkulov proved an analogous $C^2$ Lusin property for subharmonic functions.
The following fundamental property of subharmonic functions
was proved by S.A.Imomkulov [4] in 1992:

Theorem. Let $f(x)$ be a subharmonic function on a domain
  $D\subset\mathbb{R}^n$. Then for any $\varepsilon>0$ there is $g\in
 C^2(\mathbb{R}^n)$ such that $ |\{x\in D:\ f(x)\neq g(x)\}|<\varepsilon. $

Unfortunately, the result is not known. According to MathSciNet the paper of Imomkulov has zero citations. Recently, another proof has been obtained in [2] although the authors were not aware of the result of Imomkulov. 
Convex functions are subharmonic and in that special case the above result was proved in [1] and in [3]. Note that both of the papers were published after the paper of Imomkulov.
[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differential Equations 2 (1994), 17–27.  
[2] G. Alberti, S. Bianchini, C. G. Stefano; Crippa, 
On the $L^p$-differentiability of certain classes of functions. Rev. Mat. Iberoam. 30 (2014), no. 1, 349–367.
[3] L. C. Evans, W. Gangbo, W. Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999), no. 653
[4] S. A. Imomkulov,
Twice differentiability of subharmonic functions. (Russian. Russian summary) 
Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 4, 877--888; translation in 
Russian Acad. Sci. Izv. Math. 41 (1993), no. 1, 157–167 
A: The following is an example from computability theory (more specifically, $\lambda$-calculus) so maybe it's a bit at the border of mathematics proper, but both the question and the answer are so simple and natural that I think it deserves to be mentioned.
We consider the simply-typed $\lambda$-calculus, with types $A,B,\ldots$ generated by a single atom $o$:
$$A,B ::= o\mathrel{|} A\to B.$$
Let $A$ be an arbitrary type and let
$$
\begin{array}{rcl}
\mathsf{Str}[A]&:=& (A\to A)\to(A\to A)\to A\to A, \\
\mathsf{Bool}&:=& o\to o\to o.
\end{array}
$$
These are the standard types of Church binary strings and booleans, respectively. By well-known results, for all $A$, every closed term $M:\mathsf{Str}[A]\to\mathsf{Bool}$ must decide some language $L\subseteq\{0,1\}^\ast$. Let $\mathsf{ST}\lambda$ be the class of such languages, i.e.,
$$\mathsf{ST}\lambda:=\{L\subseteq\{0,1\}^\ast\mathrel{|}L\text{ is decidable by some }M:\mathsf{Str}[A]\to\mathsf{Bool}\text{ for some type }A\}.$$
It is natural to ask the following:

Question: Does $\mathsf{ST}\lambda$ correspond to a well-known class?

Surprisingly, very few people know the answer. I myself mentioned in passing this question in front of various audiences of $\lambda$-calculus experts, explicitly saying that a precise characterization of $\mathsf{ST}\lambda$ was missing and that probably this did not correspond to anything remarkable (I even wrote this in a CS Theory StackExchange answer). I never got any reaction. So you can imagine how surprised I was when, a few days ago, after discussing this with two colleagues and realizing that the answer might be the well-known class of regular languages, we found out that we were right and that this had actually been known for more than 20 years:

Theorem. $\mathsf{ST}\lambda=\mathsf{REG}$ (the regular languages on $\{0,1\}$).

This is Theorem 3.4 in the following paper (which is as nice as is unknown):
Gerd G. Hillebrand, Paris C. Kanellakis:
On the Expressive Power of Simply Typed and Let-Polymorphic Lambda Calculi. LICS 1996: 253-263.
Very roughly, the proof is based on the fact that the simply-typed $\lambda$-calculus may be interpreted in the category of finite sets (it is cartesian closed) and from this interpretation one may build a finite state automaton for the language decided by a simply-typed $\lambda$-term.
Granted, this is nothing ground-shattering, researchers in the theory of $\lambda$-calculus and programming languages may survive very well (as they do) without knowing it, but I find it doubly suprising that this result is never mentioned: on the one hand, because $\mathsf{ST}\lambda$ ends up being such a simple and universally known class; on the other hand, because this is quite unexpected.  Indeed, the class of functions on natural numbers computable by simply-typed $\lambda$-terms is, depending on the convention, either the polynomials with if-then-else (an old result of Schwichtenberg) or a weird subclass of the elementary functions, which includes all towers of exponentials but fails to include, for instance, subtraction (an old result of Statman; if I remember correctly, this class was studied thoroughly by Thierry Joly in his thesis). In both cases, this seems to give the simply-typed $\lambda$-calculus far more power than deciding regular languages.
Edit (22 Nov 2021): I'd like to add that, since I wrote this answer, the above theorem of Hillebrand and Kanellakis has become the starting point of a rich theory exploring the connections between automata and $\lambda$-calculi, showing that the result is more than just a curiosity.  An account of this line of research may be found in Lê Thành Dũng "Tito" Nguyễn's Ph.D. thesis.
A: I am not sure whether the following satisfies the OP's high standards for a good answer, but I thought the result was very interesting when I first learned about it a few years ago.

Theorem. Let $E \subset \mathbb{C}$ be compact, and let $f$ be a bounded continuous function on the Riemann sphere $\widehat{\mathbb{C}}$ which is analytic on $\widehat{\mathbb{C}} \setminus E$. Then
  $$f(E)=f(\widehat{\mathbb{C}}).$$

$\phantom{a}$
I am not sure where this result was first proved. The only reference I know is the book of A.Browder, 
Introduction to Function Algebras, Lemma 3.5.4, p.199.
The proof is not difficult. If I recall correctly, it goes like this :
Proof. We want to show that for $w \in \widehat{\mathbb{C}}$, if there exists $z \in \widehat{\mathbb{C}}$ with $f(z)=w$, then there exists $z \in E$ with $f(z)=w$. Replacing $f$ by $f-w$ if necessary (or by $1/f$ if $w=\infty$), it suffices to show that if $f$ has a zero in $\widehat{\mathbb{C}}$, then it has a zero on $E$. Suppose for a contradiction that $f$ has a zero in $\widehat{\mathbb{C}}$ but does not vanish on $E$. Without loss of generality $\infty\notin E$. Then $f$ has finitely many complex zeros, say $z_1,\dots,z_n$, listed with multiplicities. Let $$g(z)=\frac{f(z)}{(z-z_1)\cdots (z-z_n)}.$$ Then $g$ is continuous and non-vanishing on $\mathbb{C}$ and $g(\infty)=0$. So there is a continuous function $h$ on $\mathbb{C}$ analytic outside $E\cup \infty$ with $g=e^h$. But this is impossible $g$ has a pole of finite order at $\infty$.
Now, as an example :

Example. Let $\Gamma \subset \mathbb{C}$ be a curve with Hausdorff dimension bigger than one. Then by Frostman's lemma, there is a nontrivial Radon measure supported on $E$ with growth $\mu(\mathbb{D}(z_0,r)) \leq r^{1+\epsilon}$ for all $z_0 \in \mathbb{C}$, $r>0$, for some small $\epsilon>0$. The growth condition on $\mu$ implies that the Cauchy transform
  $$f(z) = \int_\Gamma \frac{d\mu(\zeta)}{\zeta-z} \qquad (z \in \widehat{\mathbb{C}}\setminus E)$$
  is Holder continuous (see Theorem 2.10 in M. Younsi, On removable sets for holomorphic functions, EMS Surv. Math. Sci. 2 (2015), no. 2, 219-254.), hence in particular extends to be continuous on the whole sphere. By the theorem, the function $f$ maps $\Gamma$ to a space-filling curve!

A: 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.
A: The Lévy Continuity Theorem for random Schwartz distributions due to Fernique: 
First let me recall the well known Lévy continuity theorem for Borel probability measures on a finite-dimensional vector space $V$ like $\mathbb{R}^d$.
To a probability measure $\mu$ one can associate the characteristic function
$$
\begin{array}{llll}
\Phi_{\mu}: & V' & \longrightarrow & \mathbb{C}\\
  & \ell & \longmapsto & \Phi_{\mu}(\ell)= \int_V e^{i \ell(v)}\ d\mu(v)
\end{array}
$$
which is defined on the dual space.
By definition, a sequence of probability measures $(\mu_n)$ converges weakly to a probability measure $\mu$, iff for all bounded continuous functions $F:V\rightarrow \mathbb{R}$ (or $\mathbb{C}$),
$$
\lim_{n\rightarrow \infty} \int_V F(v)\ d\mu_n(v) =  \int_V F(v)\ d\mu(v)\ .
$$
For $V$-valued random variables, this corresponds to convergence in law or in distribution.
We now have the following well-known result say for $V=\mathbb{R}^d$.

Lévy Continuity Theorem:
A sequence of Borel probability measures $(\mu_n)$ on $V$ converges weakly to some (unspecified) Borel probability measure iff the corresponding characteristic functions $\Phi_{\mu_n}$ converge pointwise on $V'$ to some function which is continuous at the origin.

Now the not well known result I propose in this answer, is the analogue for $V=\mathcal{S}'(\mathbb{R}^d)$ the space of temperate Schwartz distributions on $\mathbb{R}^d$.

Lévy-Fernique Continuity Theorem:
A sequence of Borel probability measures $(\mu_n)$ on $V$ converges weakly to some (unspecified) Borel probability measure iff the corresponding characteristic functions $\Phi_{\mu_n}$ converge pointwise on $V'$ to some function which is continuous at the origin.

To clarify, here $V=\mathcal{S}'(\mathbb{R}^d)$ equipped with the strong topology. Also $V'$ is the topological dual equipped with the strong topology. One has $V'\simeq \mathcal{S}(\mathbb{R}^d)$ with its usual topology, i.e., these spaces are reflexive. The definitions of characteristic functions and weak convergence of Borel probability measures are the same as in the finite-dimensional case above.
Comments:
This is important because just about any random object/process can be seen as living inside a space of distributions like $\mathcal{S}'$ or $\mathcal{D}'$.
(Will add more comments later when I find time).
References:


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*X. Fernique, Processus linéaires, processus généralisés,
Annales de l'Institut Fourier, Volume 17 (1967) no. 1, p. 1-92. 

*H. Biermé, O. Durieu, Y. Wang, Generalized random fields and Lévy's continuity theorem on the space of tempered distributions, arXiv 2017.

