Conull set

Last updated March 21, 2019

In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero.^[1] For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure.^[2]

Complement (set theory) complement of a set A in a set C is the set that includes all C-elements that are not in A and no A element, if A is included in C

In set theory, the complement of a set $A$ refers to elements not in $A$ .

In mathematical analysis, a null set $is a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set in set theory anticipates the development of Lebesgue measure since a null set necessarily has measure zero . More generally, on a given measure space a null set is a set such that .$

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the $n$ -dimensional Euclidean space $R n$ . For instance, the Lebesgue measure of the interval $[0, 1]$ in the real numbers is its length in the everyday sense of the word, specifically, 1.

A property that is true of the elements of a conull set is said to be true almost everywhere.^[3]

In measure theory, a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of almost everywhere is a companion notion to the concept of measure zero. In the subject of probability, which is largely based in measure theory, the notion is referred to as almost surely.

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References

↑ Führ, Hartmut (2005), Abstract harmonic analysis of continuous wavelet transforms, Lecture Notes in Mathematics, 1863, Springer-Verlag, Berlin, p. 12, ISBN 3-540-24259-7, MR 2130226 .
↑ A related but slightly more complex example is given by Führ, p. 143.
↑ Bezuglyi, Sergey (2000), "Groups of automorphisms of a measure space and weak equivalence of cocycles", Descriptive set theory and dynamical systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., 277, Cambridge Univ. Press, Cambridge, pp. 59–86, MR 1774424 . See p. 62 for an example of this usage.

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[1] Führ, Hartmut (2005), Abstract harmonic analysis of continuous wavelet transforms, Lecture Notes in Mathematics, 1863, Springer-Verlag, Berlin, p. 12, ISBN 3-540-24259-7, MR 2130226 .

[2] A related but slightly more complex example is given by Führ, p. 143.

[3] Bezuglyi, Sergey (2000), "Groups of automorphisms of a measure space and weak equivalence of cocycles", Descriptive set theory and dynamical systems (Marseille-Luminy, 1996), London Math. Soc. Lecture Note Ser., 277, Cambridge Univ. Press, Cambridge, pp. 59–86, MR 1774424 . See p. 62 for an example of this usage.