uniform convergence

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In the
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
, uniform convergence is a mode of
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four We ...
of functions stronger than
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
. A
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of functions $\left(f_n\right)$ converges uniformly to a limiting function $f$ on a set $E$ if, given any arbitrarily small positive number $\epsilon$, a number $N$ can be found such that each of the functions $f_N, f_,f_,\ldots$ differs from $f$ by no more than $\epsilon$ ''at every point'' $x$ ''in'' $E$. Described in an informal way, if $f_n$ converges to $f$ uniformly, then the rate at which $f_n\left(x\right)$ approaches $f\left(x\right)$ is "uniform" throughout its domain in the following sense: in order to guarantee that $f_n\left(x\right)$ falls within a certain distance $\epsilon$ of $f\left(x\right)$, we do not need to know the value of $x\in E$ in question — there can be found a single value of $N=N\left(\epsilon\right)$ ''independent of $x$'', such that choosing $n\geq N$ will ensure that $f_n\left(x\right)$ is within $\epsilon$ of $f\left(x\right)$ ''for all $x\in E$''. In contrast, pointwise convergence of $f_n$ to $f$ merely guarantees that for any $x\in E$ given in advance, we can find $N=N\left(\epsilon, x\right)$ ($N$ can depend on the value of ''$x$'') so that, ''for that particular'' ''$x$'', $f_n\left(x\right)$ falls within $\epsilon$ of $f\left(x\right)$ whenever $n\geq N$. The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, is important because several properties of the functions $f_n$, such as continuity, Riemann integrability, and, with additional hypotheses,
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, are transferred to the limit $f$ if the convergence is uniform, but not necessarily if the convergence is not uniform.

# History

In 1821
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
published a proof that a convergent sum of continuous functions is always continuous, to which
Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
in 1826 found purported counterexamples in the context of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions. The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series $\sum_^\infty f_n(x,\phi,\psi)$ is independent of the variables $\phi$ and $\psi.$ While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs. Later Gudermann's pupil
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, who attended his course on elliptic functions in 1839–1840, coined the term ''gleichmäßig konvergent'' (german: uniformly convergent) which he used in his 1841 paper ''Zur Theorie der Potenzreihen'', published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel and
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University of Cambridge, where he was the Lu ...
. G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis." Under the influence of Weierstrass and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

# Definition

We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below). Suppose $E$ is a set and $\left(f_n\right)_$ is a sequence of real-valued functions on it. We say the sequence $\left(f_n\right)_$ is uniformly convergent on $E$ with limit $f: E \to \R$ if for every $\epsilon > 0,$ there exists a natural number $N$ such that for all $n \geq N$ and for all $x \in E$ :$, f_n\left(x\right)-f\left(x\right), <\epsilon.$ The notation for uniform convergence of $f_n$ to $f$ is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity): :$f_n\rightrightarrows f, \quad \undersetf_n = f, \quad f_n \overset f, \quad f=u-\lim_ f_n .$ Frequently, no special symbol is used, and authors simply write :$f_n\to f \quad \mathrm$ to indicate that convergence is uniform. (In contrast, the expression $f_n\to f$ on $E$ without an adverb is taken to mean
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
on $E$: for all $x \in E$, $f_n\left(x\right)\to f\left(x\right)$ as $n\to\infty$.) Since $\R$ is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: $\left(f_n\right)_$ converges uniformly on $E$ (in the previous sense) if and only if for every $\epsilon > 0$, there exists a natural number $N$ such that :$x\in E, m,n\geq N \implies , f_m\left(x\right)-f_n\left(x\right), <\epsilon$. In yet another equivalent formulation, if we define :$d_n = \sup_ , f_n\left(x\right) - f\left(x\right) , ,$ then $f_n$ converges to $f$ uniformly if and only if $d_n\to 0$ as $n\to\infty$. Thus, we can characterize uniform convergence of $\left(f_n\right)_$ on $E$ as (simple) convergence of $\left(f_n\right)_$ in the function space $\R^E$ with respect to the '' uniform metric'' (also called the supremum metric), defined by :$d\left(f,g\right)=\sup_ , f\left(x\right)-g\left(x\right), .$ Symbolically, :$f_n\rightrightarrows f\iff \lim_ d\left(f_n,f\right)= 0$. The sequence $\left(f_n\right)_$ is said to be locally uniformly convergent with limit $f$ if $E$ is a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
and for every $x\in E$, there exists an $r > 0$ such that $\left(f_n\right)$ converges uniformly on $B\left(x,r\right)\cap E.$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.

## Notes

Intuitively, a sequence of functions $f_n$ converges uniformly to $f$ if, given an arbitrarily small $\epsilon>0$, we can find an $N\in\N$ so that the functions $f_n$ with $n>N$ all fall within a "tube" of width $2\epsilon$ centered around $f$ (i.e., between $f\left(x\right)-\epsilon$ and $f\left(x\right)+\epsilon$) for the ''entire domain'' of the function. Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all $x\in E$" in front of "there exists a natural number $N$" results in a definition of
pointwise convergence In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. Definition Suppose that X is a set and ...
of the sequence. To make this difference explicit, in the case of uniform convergence, $N=N\left(\epsilon\right)$ can only depend on $\epsilon$, and the choice of $N$ has to work for all $x\in E$, for a specific value of $\epsilon$ that is given. In contrast, in the case of pointwise convergence, $N=N\left(\epsilon,x\right)$ may depend on both $\epsilon$ and $x$, and the choice of $N$ only has to work for the specific values of $\epsilon$ and $x$ that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates.

## Generalizations

One may straightforwardly extend the concept to functions ''E'' → ''M'', where (''M'', ''d'') is a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, by replacing $, f_n\left(x\right)-f\left(x\right),$ with $d\left(f_n\left(x\right),f\left(x\right)\right)$. The most general setting is the uniform convergence of nets of functions ''E'' → ''X'', where ''X'' is a
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uni ...
. We say that the net $\left(f_\alpha\right)$ ''converges uniformly'' with limit ''f'' : ''E'' → ''X'' if and only if for every
entourage An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to: Arts and entertainment * L'entourage, French hip hop / rap collect ...
''V'' in ''X'', there exists an $\alpha_0$, such that for every ''x'' in ''E'' and every $\alpha\geq \alpha_0$, $\left(f_\alpha\left(x\right),f\left(x\right)\right)$ is in ''V''. In this situation, uniform limit of continuous functions remains continuous.

## Definition in a hyperreal setting

Uniform convergence admits a simplified definition in a hyperreal setting. Thus, a sequence $f_n$ converges to ''f'' uniformly if for all ''x'' in the domain of $f^*$ and all infinite ''n'', $f_n^*\left(x\right)$ is infinitely close to $f^*\left(x\right)$ (see
microcontinuity In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is i ...
for a similar definition of uniform continuity).

# Examples

For $x \in \left[0,1\right)$, a basic example of uniform convergence can be illustrated as follows: the sequence $\left(1/2\right)^$ converges uniformly, while $x^n$ does not. Specifically, assume $\epsilon=1/4$. Each function $\left(1/2\right)^$ is less than or equal to $1/4$ when $n \geq 2$, regardless of the value of $x$. On the other hand, $x^n$ is only less than or equal to $1/4$ at ever increasing values of $n$ when values of $x$ are selected closer and closer to 1 (explained more in depth further below). Given a topological space ''X'', we can equip the space of bounded function, bounded real number, real or complex number, complex-valued functions over ''X'' with the uniform norm topology, with the uniform metric defined by :$d\left(f,g\right)=\, f-g\, _=\sup_ , f\left(x\right)-g\left(x\right), .$ Then uniform convergence simply means
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen * "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that united the four We ...
in the uniform norm topology: :$\lim_\, f_n-f\, _=0$. The sequence of functions $\left(f_n\right)$ : is a classic example of a sequence of functions that converges to a function $f$ pointwise but not uniformly. To show this, we first observe that the pointwise limit of $\left(f_n\right)$ as $n\to\infty$ is the function $f$, given by : $f\left(x\right) = \lim_ f_n\left(x\right) = \begin 0, & x \in \left[0,1\right); \\ 1, & x=1. \end$ ''Pointwise convergence:'' Convergence is trivial for $x=0$ and $x=1$, since $f_n\left(0\right)=f\left(0\right)=0$ and $f_n\left(1\right)=f\left(1\right)=1$, for all $n$. For $x \in \left(0,1\right)$ and given $\epsilon>0$, we can ensure that $, f_n\left(x\right)-f\left(x\right), <\epsilon$ whenever $n\geq N$ by choosing $N = \lceil\log\epsilon/\log x\rceil$ (here the upper square brackets indicate rounding up, see Floor and ceiling functions, ceiling function). Hence, $f_n\to f$ pointwise for all

## Exponential function

The series expansion of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
can be shown to be uniformly convergent on any bounded subset $S \subset \C$ using the Weierstrass M-test. Theorem (Weierstrass M-test). ''Let $\left(f_n\right)$ be a sequence of functions $f_n:E\to \C$ and let $M_n$ be a sequence of positive real numbers such that $, f_n\left(x\right), \le M_n$ for all $x\in E$ and $n=1,2, 3, \ldots$ If $\sum_n M_n$ converges, then $\sum_n f_n$ converges uniformly on $E$.'' The complex exponential function can be expressed as the series: :$\sum_^\frac.$ Any bounded subset is a subset of some disc $D_R$ of radius $R,$ centered on the origin in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. The Weierstrass M-test requires us to find an upper bound $M_n$ on the terms of the series, with $M_n$ independent of the position in the disc: :$\left, \frac \\le M_n, \forall z\in D_R.$ To do this, we notice :$\left, \frac\ \le \frac \le \frac$ and take $M_n=\tfrac.$ If $\sum_^M_n$ is convergent, then the M-test asserts that the original series is uniformly convergent. The ratio test can be used here: :$\lim_\frac=\lim_\frac\frac=\lim_\frac=0$ which means the series over $M_n$ is convergent. Thus the original series converges uniformly for all $z\in D_R,$ and since $S\subset D_R$, the series is also uniformly convergent on $S.$

# Properties

* Every uniformly convergent sequence is locally uniformly convergent. * Every locally uniformly convergent sequence is compactly convergent. * For
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
s local uniform convergence and compact convergence coincide. * A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy. * If $S$ is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
interval (or in general a compact topological space), and $\left(f_n\right)$ is a monotone increasing sequence (meaning $f_n\left(x\right) \leq f_\left(x\right)$ for all ''n'' and ''x'') of ''continuous'' functions with a pointwise limit $f$ which is also continuous, then the convergence is necessarily uniform ( Dini's theorem). Uniform convergence is also guaranteed if $S$ is a compact interval and $\left(f_n\right)$ is an
equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable f ...
sequence that converges pointwise.

# Applications

## To continuity

If $E$ and $M$ are topological spaces, then it makes sense to talk about the continuity of the functions $f_n,f:E\to M$. If we further assume that $M$ is a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, then (uniform) convergence of the $f_n$ to $f$ is also well defined. The following result states that continuity is preserved by uniform convergence: This theorem is proved by the " trick", and is the archetypal example of this trick: to prove a given inequality (), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (), and then combines them via the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
to produce the desired inequality. This theorem is an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous (originally stated in terms of convergent series of continuous functions) is infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. More precisely, this theorem states that the uniform limit of ''
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
'' functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.

## To differentiability

If $S$ is an interval and all the functions $f_n$ are
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point i ...
and converge to a limit $f$, it is often desirable to determine the derivative function $f\text{'}$ by taking the limit of the sequence $f\text{'}_n$. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable (not even if the sequence consists of everywhere- analytic functions, see
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstras ...
), and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance $f_n\left(x\right) = n^$ with uniform limit $f_n\rightrightarrows f\equiv 0$. Clearly, $f\text{'}$ is also identically zero. However, the derivatives of the sequence of functions are given by $f\text{'}_n\left(x\right)=n^\cos nx,$ and the sequence $f\text{'}_n$ does not converge to $f\text{'},$ or even to any function at all. In order to ensure a connection between the limit of a sequence of differentiable functions and the limit of the sequence of derivatives, the uniform convergence of the sequence of derivatives plus the convergence of the sequence of functions at at least one point is required:Rudin, Walter (1976). '' Principles of Mathematical Analysis'' 3rd edition, Theorem 7.17. McGraw-Hill: New York. : ''If $\left(f_n\right)$ is a sequence of differentiable functions on

## To integrability

Similarly, one often wants to exchange integrals and limit processes. For the
Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
, this can be done if uniform convergence is assumed: : ''If $\left(f_n\right)_^\infty$ is a sequence of Riemann integrable functions defined on a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
interval $I$ which uniformly converge with limit $f$, then $f$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $f_n$:'' $\int_I f = \lim_\int_I f_n.$ In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for ''n'' sufficiently large, the graph of $f_n$ is within of the graph of ''f'', and so the upper sum and lower sum of $f_n$ are each within $\varepsilon , I,$ of the value of the upper and lower sums of $f$, respectively. Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...

## To analyticity

Using Morera's Theorem, one can show that if a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see
Weierstrass function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstras ...
).

## To series

We say that $\sum_^\infty f_n$ converges: With this definition comes the following result:
Let ''x''0 be contained in the set ''E'' and each ''f''''n'' be continuous at ''x''0. If $f = \sum_^\infty f_n$ converges uniformly on ''E'' then ''f'' is continuous at ''x''0 in ''E''. Suppose that ''n'' is integrable on ''E''. If $\sum_^\infty f_n$ converges uniformly on ''E'' then ''f'' is integrable on ''E'' and the series of integrals of ''f''''n'' is equal to integral of the series of fn.

# Almost uniform convergence

If the domain of the functions is a measure space ''E'' then the related notion of almost uniform convergence can be defined. We say a sequence of functions $\left(f_n\right)$ converges almost uniformly on ''E'' if for every $\delta > 0$ there exists a measurable set $E_\delta$ with measure less than $\delta$ such that the sequence of functions $\left(f_n\right)$ converges uniformly on $E \setminus E_\delta$. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement. Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
as might be inferred from the name. However, Egorov's theorem does guarantee that on a finite measure space, a sequence of functions that converges
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
also converges almost uniformly on the same set. Almost uniform convergence implies almost everywhere convergence and convergence in measure.

* Uniform convergence in probability *
Modes of convergence (annotated index) The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of converge ...
* Dini's theorem * Arzelà–Ascoli theorem

# References

* Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, . * G. H. Hardy, Sir George Stokes and the concept of uniform convergence; Proceedings of the Cambridge Philosophical Society, 19, pp. 148–156 (1918) * Bourbaki; Elements of Mathematics: General Topology. Chapters 5–10 (paperback); * Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw–Hill, 1976. * Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, . * William Wade, An Introduction to Analysis, 3rd ed., Pearson, 2005