Sigma-ring

In mathematics, a nonempty collection of sets $mathcal\{R\}$ is called a σ-ring (pronounced "sigma-ring") if it is closed under countable union and relative complementation:
# if $A\_\{n\}\; in\; mathcal\{R\}$ for all $n\; in\; mathbb\{N\}$
#$A\; -\; B\; in\; mathcal\{R\}$ if $A,\; B\; in\; mathcal\{R\}$

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

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

See also

*Delta ring
*Ring of sets
*Sigma field

References

* Walter Rudin, 1976. "Principles of Mathematical Analysis", 3rd. ed. McGraw-Hill. Final chapter uses σ-rings in development of Lebesgue theory.