# 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 .

## Individual evidence

1. 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 .).${\ displaystyle C ^ {\ infty}}$
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 .
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 .