Modal companion

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

1. $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in ℛ$ if $A_n\in ℛ$ for all $n\in \mathbb{N}$
2. $A\setminus B\in ℛ$ if $A,B\in ℛ$

From these two properties we immediately see that

$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n\in ℛ$ if $A_n\in ℛ$ for all $n\in \mathbb{N}$

This is simply because ${\cap }_{n=1}^{\mathrm{\infty }}{A}_n=A_1\setminus {\cup }_{n=1}^{\mathrm{\infty }}(A_1\setminus A_n)$.

If the first property is weakened to closure under finite union (i.e., $A\cup B\in ℛ$ whenever $A,B\in ℛ$) but not countable union, then $ℛ$ 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.

A σ-ring induces a σ-field. If $ℛ$ is a σ-ring over the set $X$, then define $𝒜$ to be the collection of all subsets of X that are elements of $ℛ$ or whose complements are elements of $ℛ$. We see that $𝒜$ is a σ-field over the set X.

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.