Luzin's theorem
Encyclopedia
In mathematics
, Lusin's theorem (more properly Luzin's theorem, named for Nikolai Luzin
) in real analysis
is a form of Littlewood's second principle.
It states that every measurable function
is a continuous function
on nearly all its domain:
For an interval [a, b], let
be a measurable function. Then given
, there exists a compact 
such that ƒ restricted to E is continuous and
Here Ec denotes the complement of E. Note that E inherits the subspace topology from [a, b]; continuity of ƒ restricted to E is defined using this topology.
in L1([a, b]), there exists a sequence of continuous functions gn tending to ƒ in the L1 norm. Passing to a subsequence if necessary, we may also assume that gn tends to ƒ almost everywhere
. By Egorov's theorem
, it follows that gn tends to ƒ uniformly off some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Lusin's theorem (more properly Luzin's theorem, named for Nikolai Luzin
Nikolai Luzin
Nikolai Nikolaevich Luzin, , was a Soviet/Russian mathematician known for his work in descriptive set theory and aspects of mathematical analysis with strong connections to point-set topology. He was the eponym of Luzitania, a loose group of young Moscow mathematicians of the first half of the...
) in real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
is a form of Littlewood's second principle.
It states that every measurable function
Measurable function
In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration...
is a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
on nearly all its domain:
For an interval [a, b], let
be a measurable function. Then given
, there exists a compact such that ƒ restricted to E is continuous andHere Ec denotes the complement of E. Note that E inherits the subspace topology from [a, b]; continuity of ƒ restricted to E is defined using this topology.
A proof of Lusin's theorem
Since ƒ is measurable, it is bounded on the complement of some open set of arbitrarily small measure. So, redefining ƒ to be 0 on this open set if necessary, we may assume that ƒ is bounded and hence integrable. Since continuous functions are denseDense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in L1([a, b]), there exists a sequence of continuous functions gn tending to ƒ in the L1 norm. Passing to a subsequence if necessary, we may also assume that gn tends to ƒ almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
. By Egorov's theorem
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 follows that gn tends to ƒ uniformly off some open set of arbitrarily small measure. Since uniform limits of continuous functions are continuous, the theorem is proved.