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
:
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
:
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
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
be a measurable function. Given
, for every
of finite measure there is a closed set
with
such that
restricted to
is continuous.
(If
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
, we can choose
to be compact and even find a continuous function
with compact support that coincides with
on
and such that
:
.)
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 ...
on the unit interval