Luzin's Theorem
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In the
mathematical 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 ...
field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, Lusin's theorem (or Luzin's theorem, named for
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 ...
) or Lusin's criterion states that an almost-everywhere finite function is
measurable 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 ...
if and only if it is a
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 ...
on nearly all its domain. In the informal formulation of J. E. Littlewood, "every measurable function is nearly continuous".


Classical statement

For an interval 'a'', ''b'' let :f: ,brightarrow \mathbb be a measurable function. Then, for every ''ε'' > 0, there exists a compact ''E'' ⊆  'a'', ''b''such that ''f'' restricted to ''E'' is continuous and :\mu ( E ) > b - a - \varepsilon. Note that ''E'' inherits the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
from 'a'', ''b'' continuity of ''f'' restricted to ''E'' is defined using this topology. Also for any function ''f'', defined on the interval 'a, b''and almost-everywhere finite, if for any ''ε > 0'' there is a function ''ϕ'', continuous on 'a, b'' such that the measure of the set :\ is less than ''ε'', then ''f'' is measurable.


General form

Let (X,\Sigma,\mu) be a
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and ...
space and ''Y'' be a
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
topological space equipped with a
Borel algebra In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are ...
, and let f: X \rightarrow Y be a measurable function. Given \varepsilon>0, for every A\in\Sigma of finite measure there is a closed set E with \mu(A\setminus E) <\varepsilon such that f restricted to E is continuous. (If A is
locally compact 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 e ...
and Y=\mathbb^d, we can choose E to be compact and even find a continuous function f_\varepsilon: X \rightarrow \mathbb^d with compact support that coincides with f on E and such that :\ \sup_ , f_\varepsilon (x) , \leq \sup_ , f(x) , .) Informally, measurable functions into spaces with countable base can be approximated by continuous functions on arbitrarily large portion of their domain.


On the proof

The proof of Lusin's theorem can be found in many classical books. Intuitively, one expects it as a consequence of Egorov's theorem and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.


Example

The strength of Lusin's theorem might not be readily apparent, as can be demonstrated by example. Consider
Dirichlet function In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number). \mathb ...
, that is the
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 ...
1_\mathbb: ,1to \ on the unit interval ,1/math> taking the value of one on the rationals, and zero, otherwise. Clearly the measure of this function should be zero, but how can one find regions that are continuous, given that the rationals are
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in the reals? The requirements for Lusin's theorem can be satisfied with the following construction of a set E. Let \ be any enumeration of \mathbb. Set :G_n=(x_n-\varepsilon/2^n,x_n+\varepsilon/2^n) and :E:= ,1setminus\bigcup_^\infty G_n. Then the sequence of open sets G_n "knocks out" all of the rationals, leaving behind a compact, closed set E which contains no rationals, and has a measure of more than 1-2\varepsilon.


References

Sources * N. Lusin. Sur les propriétés des fonctions mesurables, ''Comptes rendus de l'Académie des Sciences de Paris'' 154 (1912), 1688–1690. * G. Folland. ''Real Analysis: Modern Techniques and Their Applications'', 2nd ed. Chapter 7 * W. Zygmunt. Scorza-Dragoni property (in Polish), UMCS, Lublin, 1990 * M. B. Feldman, "A Proof of Lusin's Theorem", American Math. Monthly, 88 (1981), 191-2 * Lawrence C. Evans, Ronald F. Gariepy, "Measure Theory and fine properties of functions", CRC Press Taylor & Francis Group, Textbooks in mathematics, Theorem 1.14 Citations {{reflist Theorems in real analysis Theorems in measure theory Articles containing proofs