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In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, an area of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Egorov's theorem establishes a condition for the
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
of a pointwise convergent
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 cal ...
of
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, after Carlo Severini, an
Italia Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
n
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, and
Dmitri Egorov Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Soc ...
, a
Russia Russia, or the Russian Federation, is a country spanning Eastern Europe and North Asia. It is the list of countries and dependencies by area, largest country in the world, and extends across Time in Russia, eleven time zones, sharing Borders ...
n mathematician and geometer, who published independent proofs respectively in 1910 and 1911. Egorov's theorem can be used along with
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set ...
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s to prove Lusin's theorem for
integrable function In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Inte ...
s.


Historical note

The first proof of the theorem was given by Carlo Severini in 1910: he used the result as a tool in his research on
series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used i ...
of
orthogonal functions In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval (mathematics), interval as the domain of a function, domain, the bilinear form may be the ...
. His work remained apparently unnoticed outside
Italy Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
, probably due to the fact that it is written in
Italian Italian(s) may refer to: * Anything of, from, or related to the people of Italy over the centuries ** Italians, a Romance ethnic group related to or simply a citizen of the Italian Republic or Italian Kingdom ** Italian language, a Romance languag ...
, appeared in a scientific journal with limited diffusion and was considered only as a means to obtain other theorems. A year later
Dmitri Egorov Dmitri Fyodorovich Egorov (; December 22, 1869 – September 10, 1931) was a Russian and Soviet mathematician known for contributions to the areas of differential geometry and mathematical analysis. He was President of the Moscow Mathematical Soc ...
published his independently proved results, and the theorem became widely known under his name: however, it is not uncommon to find references to this theorem as the Severini–Egoroff theorem. The first mathematicians to prove independently the theorem in the nowadays common abstract
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
setting were , and in : an earlier generalization is due to
Nikolai Luzin Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ...
, who succeeded in slightly relaxing the requirement of finiteness of measure of the
domain A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...
of convergence of the pointwise converging functions in the ample paper .According to . Further generalizations were given much later by Pavel Korovkin, in the paper , and by Gabriel Mokobodzki in the paper .


Formal statement and proof


Statement

Let (f_n) be a sequence of M-valued measurable functions, where M is a separable metric space, on some
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(X,\sigma,\mu), and suppose there is a
measurable subset In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have ...
A \subseteq X, with finite \mu-measure, such that (f_n) converges \mu-
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 to ...
on A to a limit function f. The following result holds: for every \varepsilon > 0, there exists a measurable
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
B of A such that \mu (B)<\varepsilon, and (f_n) converges to f uniformly on A\setminus B. Here, \mu (B) denotes the \mu-measure of B. In words, the theorem says that pointwise convergence almost everywhere on A implies the apparently much stronger uniform convergence everywhere except on some subset B of arbitrarily small measure. This type of convergence is called ''almost uniform convergence''.


Discussion of assumptions and a counterexample

* The hypothesis \mu (A)<\infty is necessary. To see this, it is simple to construct a counterexample when μ is the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
: consider the sequence of real-valued
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
s f_n(x) = 1_(x), \qquad n\in\N,\ x\in\R, defined on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. This sequence converges pointwise to the zero function everywhere but does not converge uniformly on \R\setminus B for any set B of finite measure: a counterexample in the general n-dimensional
real vector space Real may refer to: Currencies * Argentine real * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Nature and science * Reality, the state of things as they exist, ...
\R^n can be constructed as shown by . * The separability of the metric space is needed to make sure that for M-valued, measurable functions f and g, the distance d(f(x), g(x)) is again a measurable real-valued function of x.


Proof

Fix \varepsilon > 0. For natural numbers ''n'' and ''k'', define the set ''En,k'' by the union : E_ = \bigcup_ \left\. These sets get smaller as ''n'' increases, meaning that ''E''''n''+1,''k'' is always a subset of ''En,k'', because the first union involves fewer sets. A point ''x'', for which the sequence (''fm''(''x'')) converges to ''f''(''x''), cannot be in every ''En,k'' for a fixed ''k'', because ''fm''(''x'') has to stay closer to ''f''(''x'') than 1/''k'' eventually. Hence by the assumption of μ-almost everywhere pointwise convergence on ''A'', :\mu\left(\bigcap_E_\right)=0 for every ''k''. Since ''A'' is of finite measure, we have continuity from above; hence there exists, for each ''k'', some natural number ''nk'' such that :\mu(E_) < \frac\varepsilon. For ''x'' in this set we consider the speed of approach into the 1/''k''-
neighbourhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...
of ''f''(''x'') as too slow. Define :B = \bigcup_ E_ as the set of all those points ''x'' in ''A'', for which the speed of approach into ''at least one'' of these 1/''k''-neighbourhoods of ''f''(''x'') is too slow. On the set difference A\setminus B we therefore have uniform convergence. Explicitly, for any \epsilon, let \frac 1 k < \epsilon, then for any n > n_k, we have , f_n - f, < \epsilon on all of A\setminus B. Appealing to the
sigma additivity In mathematics, an additive set function is a function \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this ad ...
of μ and using the
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
, we get :\mu(B) \le \sum_ \mu(E_) < \sum_\frac\varepsilon=\varepsilon.


Generalizations


Luzin's version

Nikolai Luzin Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ...
's generalization of the Severini–Egorov theorem is presented here according to .


Statement

Under the same hypothesis of the abstract Severini–Egorov theorem suppose that ''A'' is the union of 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 cal ...
of
measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
s of finite μ-measure, and (''fn'') is a given sequence of ''M''-valued measurable functions on some
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(''X'',Σ,μ), such that (''f''''n'') converges μ-
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 to ...
on ''A'' to a limit function ''f'', then ''A'' can be expressed as the union of a sequence of measurable sets ''H'', ''A1'', ''A2'',... such that μ(''H'') = 0 and (''fn'') converges to ''f'' uniformly on each set ''Ak''.


Proof

It is sufficient to consider the case in which the set ''A'' is itself of finite μ-measure: using this hypothesis and the standard Severini–Egorov theorem, it is possible to define by
mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a ...
a sequence of sets k=1,2,... such that :\mu\left (A \setminus \bigcup_^ A_k \right)\leq\frac and such that (''fn'') converges to ''f'' uniformly on each set ''Ak'' for each ''k''. Choosing :H=A\setminus\bigcup_^ A_k then obviously μ(''H'') = 0 and the theorem is proved.


Korovkin's version

The proof of the Korovkin version follows closely the version on , which however generalizes it to some extent by considering
admissible functional Admissibility may refer to: Law * Admissible evidence, evidence which may be introduced in a court of law * Admissibility (ECHR), whether a case will be considered in the European Convention on Human Rights system Mathematics and logic * Admissible ...
s instead of non-negative measures and
inequalities Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of i ...
\leq and \geq respectively in conditions 1 and 2.


Statement

Let (''M'',''d'') denote a separable
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
and (''X'',Σ) a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
: consider a
measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
''A'' and a
class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...
\mathfrak containing ''A'' and its measurable
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s such that their
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
in unions and intersections belong to the same class. Suppose there exists a non-negative measure μ such that μ(''A'') exists and # \mu(\cap A_n) = \lim \mu(A_n) if A_1 \supset A_2 \supset \cdots with A_n\in\mathfrak for all ''n'' # \mu(\cup A_n) = \lim \mu(A_n) if A_1 \subset A_2 \subset \cdots with \cup A_n\in\mathfrak. If (''f''''n'') is a sequence of M-valued measurable functions converging μ-
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 to ...
on A\in\mathfrak to a limit function ''f'', then there exists a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
''A′'' of ''A'' such that 0 < μ(''A'') − μ(''A′'') < ε and where the convergence is also uniform.


Proof

Consider the indexed family of sets whose
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
is the set of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s m\in\N, defined as follows: :A_=\left\ Obviously :A_\subseteq A_\subseteq A_\subseteq\dots and :A=\bigcup_A_ therefore there is a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
''m0'' such that putting ''A0,m0''=''A0'' the following relation holds true: :0\leq\mu(A)-\mu(A_0)\leq\varepsilon Using ''A0'' it is possible to define the following indexed family :A_=\left\ satisfying the following two relationships, analogous to the previously found ones, i.e. :A_\subseteq A_\subseteq A_\subseteq\dots and :A_0=\bigcup_A_ This fact enable us to define the set ''A1,m1''=''A1'', where ''m1'' is a surely existing natural number such that :0\leq\mu(A)-\mu(A_1)\leq\varepsilon By iterating the shown construction, another indexed family of set is defined such that it has the following properties: * A_0\supseteq A_1\supseteq A_2\supseteq\cdots * 0\leq\mu(A)-\mu(A_m)\leq\varepsilon for all m\in\N * for each m\in\N there exists ''km'' such that for all n \geq k_m then d(f_n(x),f(x)) \le 2^ for all x \in A_m and finally putting :A'=\bigcup_A_n the thesis is easily proved.


Notes


References


Historical references

*, available at Gallica. *. *. *. Published by the
Accademia Gioenia Accademia (Italian for "academy") often refers to: * The Galleria dell'Accademia, an art museum in Florence * The Gallerie dell'Accademia, an art museum in Venice Accademia may also refer to: Academies of art * The Accademia Carrara di Belle ...
in
Catania Catania (, , , Sicilian and ) is the second-largest municipality on Sicily, after Palermo, both by area and by population. Despite being the second city of the island, Catania is the center of the most densely populated Sicilian conurbation, wh ...
. *. *, available from th
Biblioteca Digitale Italiana di Matematica
The
obituary An obituary (wikt:obit#Etymology 2, obit for short) is an Article (publishing), article about a recently death, deceased person. Newspapers often publish obituaries as Article (publishing), news articles. Although obituaries tend to focus on p ...
of Carlo Severini. *. A short note in which Leonida Tonelli credits Severini for the first proof of Severini–Egorov theorem.


Scientific references

* *. A definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of
sequences 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 call ...
of measure-related structures (measurable functions,
measurable set In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
s, measures and their combinations) is somewhat conclusive. *. Contains a section named ''Egorov type theorems'', where the basic Severini–Egorov theorem is given in a form which slightly generalizes that of . * * * *, reviewed by and by . * (available at th
Polish Virtual Library of Science
.


External links

* * * {{Measure theory Theorems in measure theory Articles containing proofs