Constant-weight code: Difference between revisions

Revision as of 08:42, 29 November 2013

In mathematics, a nonempty collection of sets $ℛ$ is called a δ-ring (pronounced delta-ring) if it is closed under union, relative complementation, and countable intersection:

$A \cup B \in ℛ$ if $A, B \in ℛ$
$A - B \in ℛ$ if $A, B \in ℛ$
$⋂_{n = 1}^{\infty} A_{n} \in ℛ$ if $A_{n} \in ℛ$ for all $n \in ℕ$

If only the first two properties are satisfied, then $ℛ$ is a ring but not a δ-ring. Every σ-ring is a δ-ring, but not every δ-ring is a σ-ring.

δ-rings can be used instead of σ-fields in the development of measure theory if one does not wish to allow sets of infinite measure.

References

Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html

Template:Mathanalysis-stub

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+In [[mathematics]], a nonempty collection of [[Set (mathematics)|sets]] <math>\mathcal{R}</math> is called a '''δ-ring''' (pronounced ''delta-ring'') if it is [[closure (mathematics)|closed]] under [[union (set theory)|union]], [[Complement (set theory)|relative complementation]], and countable [[Intersection (set theory)|intersection]]:
+#<math>A \cup B \in \mathcal{R}</math> if <math>A, B \in \mathcal{R}</math>
+#<math>A - B \in \mathcal{R}</math> if <math>A, B \in \mathcal{R}</math>
+#<math>\bigcap_{n=1}^{\infty} A_{n} \in \mathcal{R}</math> if <math>A_{n} \in \mathcal{R}</math> for all <math>n \in \mathbb{N}</math>
+If only the first two properties are satisfied, then <math>\mathcal{R}</math> is a [[Ring of sets|ring]] but not a δ-ring. Every [[Sigma-ring|σ-ring]] is a δ-ring, but not every δ-ring is a σ-ring.
+δ-rings can be used instead of [[Sigma-algebra|σ-fields]] in the development of [[Measure (mathematics)|measure]] theory if one does not wish to allow sets of infinite measure.
+== See also ==
+*[[Ring of sets]]
+*[[Sigma-algebra|Sigma field]]
+*[[Sigma ring]]
+== References ==
+* Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html
+{{Mathanalysis-stub}}
+[[Category:Measure theory]]
+[[Category:Set families]]

Constant-weight code: Difference between revisions

Revision as of 08:42, 29 November 2013

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