Modal companion

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:

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

From these two properties we immediately see that

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

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

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.

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