Zero quantity

In mathematics, a zero set (or zero set ) is a subset of a measure space (more precisely: is an element of the associated σ-algebra ) that has the measure zero. It is not to be confused with the empty set ; in fact, a null set can even contain an infinite number of elements. Some authors also add negligible quantities to the definition of zero set , i.e. H. those that are a subset of a null set, but not necessarily an element of algebra, and which therefore may not be assigned a measure. If a measure is also assigned to all sets that differ from an element of algebra by such a negligible amount , one speaks of the completion of the measure, as it is used in the definition of the Lebesgue measure . ${\ displaystyle \ mu}$ ${\ displaystyle A}$ ${\ displaystyle (\ Omega, \ Sigma, \ mu)}$ ${\ displaystyle A}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle \ sigma}$ ${\ displaystyle \ sigma}$

From a property that for all elements of Maßraums outside a valid -null, they say that they - almost everywhere applies. Is a probability measure, so one also says - almost certainly instead of - almost everywhere. ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu}$

Examples

The empty set forms a zero set in every measure space. ${\ displaystyle \ emptyset}$

The following applies to the Lebesgue measure on or on : ${\ displaystyle \ lambda}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ lambda _ {n}}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

A subset of is a Lebesgue null set if and only if there is a sequence of paraxial -dimensional cubes or cuboids with and . ${\ displaystyle N}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ varepsilon> 0}$ ${\ displaystyle \ left (I_ {i} \ right) _ {i \ in \ mathbb {N}}}$ ${\ displaystyle n}$ ${\ displaystyle N \ subset \ bigcup \ limits _ {i \ in \ mathbb {N}} I_ {i}}$ ${\ displaystyle \ sum \ limits _ {i \ in \ mathbb {N}} \ lambda _ {n} \ left (I_ {i} \ right) <\ varepsilon}$

Every countable subset of is a null set. In particular, the set of rational numbers in the set of real numbers is a zero set. ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {R}}$

Every real subspace , especially every hyperplane , is a null set. The same is true for affine subspaces and submanifolds whose dimension is smaller than . ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle n}$

The Cantor set is an uncountable zero set in the set of real numbers.

Generalizations

Content on half-rings

Zero sets can also be defined more generally for elements of a half-ring . A set from is called a null set if the content holds . This generalization includes both the above definition, since every algebra is also a half-ring and every measure is also a content, as well as the case for rings and premeasures . ${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle A}$ ${\ displaystyle {\ mathcal {H}}}$ ${\ displaystyle \ mu}$ ${\ displaystyle \ mu (A) = 0}$ ${\ displaystyle \ sigma}$

Differentiable manifolds

For differentiable manifolds there is generally no meaningful generalization of the Lebesgue measure. Nevertheless, the concept of Lebesgue nulls can be meaningfully transferred to differentiable manifolds: Let be a -dimensional differentiable manifold and , then a Lebesgue null set if for every card with the set is a Lebesgue null set in . ${\ displaystyle M}$ ${\ displaystyle n}$ ${\ displaystyle C \ subset M}$ ${\ displaystyle C}$ ${\ displaystyle h \;: \; U \ rightarrow V}$ ${\ displaystyle V \ subset \ mathbb {R} ^ {n}}$ ${\ displaystyle h \ left (C \ cap U \ right)}$ ${\ displaystyle \ mathbb {R} ^ {n}}$

With this definition, Sard's theorem can be applied to differentiable manifolds. In the case of pseudo-Riemannian manifolds , these Lebesgue null sets are identical to the null sets with respect to the Riemann-Lebesgue volume measure .

literature

Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin a. a. 2005, ISBN 3-540-21390-2 .
Heinz Bauer : Measure and integration theory. 2nd, revised edition. de Gruyter, Berlin a. a. 1992, ISBN 3-11-013626-0 .

Individual evidence

↑ ^a ^b Theodor Bröcker , Klaus Jänich : Introduction to the differential topology (= Heidelberger Taschenbücher . Volume 143 ). Springer Verlag, Berlin / Heidelberg / u. a. 1990, ISBN 3-540-06461-3 , § 6. The Sard Theorem , Definitions 6.1 and 6.3, pp. 58–59 (Corrected reprint. “Differentiable” is always meant here .). ${\ displaystyle C ^ {\ infty}}$
^ Herbert Amann, Joachim Escher : Analysis III . 2nd Edition. Birkhäuser Verlag, Basel 2008, ISBN 978-3-7643-8883-6 , Chapter IX. Elements of Measure Theory , 5. The Lebesgue Measure , Theorem 5.1 (v), p. 41 .
^ Herbert Amann, Joachim Escher : Analysis III . 2nd Edition. Birkhäuser Verlag, Basel 2008, ISBN 978-3-7643-8883-6 , Chapter XII. Integration on Manifolds , 1. Volume measures , Theorem 1.6, p. 409 .

[broecker_jaenich-1] Theodor Bröcker , Klaus Jänich : Introduction to the differential topology (= Heidelberger Taschenbücher . Volume 143 ). Springer Verlag, Berlin / Heidelberg / u. a. 1990, ISBN 3-540-06461-3 , § 6. The Sard Theorem , Definitions 6.1 and 6.3, pp. 58–59 (Corrected reprint. “Differentiable” is always meant here .). ${\ displaystyle C ^ {\ infty}}$

[amann_escher_lbsg-2] Herbert Amann, Joachim Escher : Analysis III . 2nd Edition. Birkhäuser Verlag, Basel 2008, ISBN 978-3-7643-8883-6 , Chapter IX. Elements of Measure Theory , 5. The Lebesgue Measure , Theorem 5.1 (v), p. 41 .

[amann_escher_mfkt-3] Herbert Amann, Joachim Escher : Analysis III . 2nd Edition. Birkhäuser Verlag, Basel 2008, ISBN 978-3-7643-8883-6 , Chapter XII. Integration on Manifolds , 1. Volume measures , Theorem 1.6, p. 409 .