Sigma-ring

Let $\backslash mathcal\{R\}$ be a nonempty collection of sets. Then $\backslash mathcal\{R\}$ is a {{sigma}}-ring if:

1. Closed under countable unions: $\backslash bigcup\_\{n=1\}^\{\backslash infty\}\; A\_\{n\}\; \backslash in\; \backslash mathcal\{R\}$ if $A\_\{n\}\; \backslash in\; \backslash mathcal\{R\}$ for all $n\; \backslash in\; \backslash N$
2. Closed under relative complementation: $A\; \backslash setminus\; B\; \backslash in\; \backslash mathcal\{R\}$ if $A,\; B\; \backslash in\; \backslash mathcal\{R\}$

These two properties imply:

$$\backslash bigcap\_\{n=1\}^\{\backslash infty\}\; A\_n\; \backslash in\; \backslash mathcal\{R\}$$

whenever $A\_1,\; A\_2,\; \backslash ldots$ are elements of $\backslash mathcal\{R\}.$

This is because

$$\backslash bigcap\_\{n=1\}^\backslash infty\; A\_n\; =\; A\_1\; \backslash setminus\; \backslash bigcup\_\{n=2\}^\{\backslash infty\}\backslash left(A\_1\; \backslash setminus\; A\_n\backslash right).$$

Every {{sigma}}-ring is a δ-ring but there exist δ-rings that are not {{sigma}}-rings.

If the first property is weakened to closure under finite union (that is, $A\; \backslash cup\; B\; \backslash in\; \backslash mathcal\{R\}$ whenever $A,\; B\; \backslash in\; \backslash mathcal\{R\}$) but not countable union, then $\backslash mathcal\{R\}$ is a ring but not a {{sigma}}-ring.

{{sigma}}-rings can be used instead of {{sigma}}-fields ({{sigma}}-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every {{sigma}}-field is also a {{sigma}}-ring, but a {{sigma}}-ring need not be a {{sigma}}-field.

A {{sigma}}-ring $\backslash mathcal\{R\}$ that is a collection of subsets of $X$ induces a {{sigma}}-field for $X.$ Define $\backslash mathcal\{A\}\; =\; \backslash \{\; E\; \backslash subseteq\; X\; :\; E\; \backslash in\; \backslash mathcal\{R\}\; \backslash \; \backslash text\{or\}\; \backslash \; E^c\; \backslash in\; \backslash mathcal\{R\}\; \backslash \}.$ Then $\backslash mathcal\{A\}$ is a {{sigma}}-field over the set $X$ - to check closure under countable union, recall a $\backslash sigma$-ring is closed under countable intersections. In fact $\backslash mathcal\{A\}$ is the minimal {{sigma}}-field containing $\backslash mathcal\{R\}$ since it must be contained in every {{sigma}}-field containing $\backslash mathcal\{R\}.$

• {{annotated link|Delta-ring|{{delta}}-ring}}
• {{annotated link|Field of sets}}
• {{annotated link|Join (sigma algebra)}}
• {{annotated link|Dynkin system|{{lambda}}-system (Dynkin system)}}
• {{annotated link|Measurable function}}
• {{annotated link|Monotone class}}
• {{annotated link|Pi-system|{{pi}}-system}}
• {{annotated link|Ring of sets}}
• {{annotated link|Sample space}}
• {{annotated link|Sigma-additive set function|{{sigma}} additivity}}
• {{annotated link|σ-algebra}}
• {{annotated link|Sigma-ideal|{{sigma}}-ideal}}

{{reflist}}

• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses {{sigma}}-rings in development of Lebesgue theory.

{{Families of sets}}

Category:Measure theory

Category:Families of sets