title: Sigma-ring
text: # Sigma-ring
  
  In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
  
  ## Formal definition
  
  Let ${\mathcal {R}}$ be a nonempty collection of sets. Then ${\mathcal {R}}$ is a 𝜎-ring if:
  
  * Closed under countable unions: $\bigcup _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ if $A_{n}\in {\mathcal {R}}$ for all $n\in \mathbb {N}$
  * Closed under relative complementation: $A\setminus B\in {\mathcal {R}}$ if $A,B\in {\mathcal {R}}$
  
  ## Properties
  
  These two properties imply:
  
  $\bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$
  
  whenever $A_{1},A_{2},\ldots$ are elements of ${\mathcal {R}}.$
  
  This is because
  
  $\bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).$
  
  Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-rings.
  
  ## Similar concepts
  
  If the first property is weakened to closure under finite union (that is, $A\cup B\in {\mathcal {R}}$ whenever $A,B\in {\mathcal {R}}$) but not countable union, then ${\mathcal {R}}$ is a ring but not a 𝜎-ring.
  
  ## Uses
  
  𝜎-rings can be used instead of 𝜎-fields (𝜎-algebras) 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 ${\mathcal {R}}$ that is a collection of subsets of $X$ induces a 𝜎-field for $X.$ Define ${\mathcal {A}}=\{E\subseteq X:E\in {\mathcal {R}}\ {\text{or}}\ E^{c}\in {\mathcal {R}}\}.$ Then ${\mathcal {A}}$ is a 𝜎-field over the set $X$ - to check closure under countable union, recall a $\sigma$-ring is closed under countable intersections. In fact ${\mathcal {A}}$ is the minimal 𝜎-field containing ${\mathcal {R}}$ since it must be contained in every 𝜎-field containing ${\mathcal {R}}.$
  
  