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 ...

, a function ''f'' on the interval 'a'', ''b''has the Luzin N property, named after

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 ...

(also called Luzin property or N property) if for all

N\subset,b /math> such that \lambda(N)=0, there holds: \lambda(f(N))=0, where \lambda stands for 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 ...

. Note that the image of such a set ''N'' is not necessarily

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 ...

, but since the Lebesgue measure is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

, it follows that if the Lebesgue

outer measure In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer me ...

of that set is zero, then it is measurable and its Lebesgue measure is zero as well.

Properties

Any differentiable function has the Luzin N property.Rudin, Real and Complex analysis
Lemma 7.25 implies this This extends to functions that are differentiable on a cocountable set, as the image of a countable set is countable and thus a null set, but not to functions differentiable on a

conull set In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure. A property that is true of the e ...

: The

Cantor function In mathematics, the Cantor function is an example of a function (mathematics), function that is continuous function, continuous, but not absolute continuity, absolutely continuous. It is a notorious Pathological_(mathematics)#Pathological_exampl ...

does not have the Luzin N property, as the Lebesgue measure of the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...

is zero, but its image is the complete ,1interval. A function ''f'' on the interval 'a'',''b''is

absolutely continuous In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship betwe ...

if and only if it is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

, is of

bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...

and has the Luzin N property.

References

External links

Luzin-N-property
in the ''

Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduat ...

'' Real analysis Measure theory {{mathanalysis-stub