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

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...

, Lusin's theorem (or Luzin's theorem, named for

Nikolai Luzin Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ...

) or Lusin's criterion states that an

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 ...

finite function is

measurable In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...

if and only if it is a

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

on nearly all its domain. In the informal formulation of

J. E. Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to mathematical analysis, analysis, number theory, and differential equations, and had lengthy collaborations with G. H. H ...

, "every measurable function is nearly continuous".

Classical statement

For an interval 'a'', ''b'' let :

rightarrow \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'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...

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 σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...

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 set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...

, 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.

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 In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions. It is also named Severini–Egoroff theorem or Severini–Egorov theorem, a ...

and density of smooth functions. Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity.

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