σ-algebra

Definition

Let $X$ be a set. A collection $\mathcal{Y}$ of subsets of $X$ is a σ-algebra over $X$ if

1. $\emptyset \in \mathcal{Y}$ (empty set in $\mathcal{Y}$),
2. $X \setminus Y \in \mathcal{Y}$ for all $Y \in \mathcal{Y}$, and
3. $\cup_{i \in \mathbb{N}} Y_i \in \mathcal{Y}$ for every sequence $(Y_i)_{i \in \mathbb{N}}$ of elements in $\mathcal{Y}$.

By De Morgan's laws, σ-algebra also closed under countable intersection.

(short version: given set $X$, σ-algebra is closed under unions and intersections of countable families of subsets)

Definition

A field $\mathcal{F}$ is a class of sets such that

1. if $A \in \mathcal{F}$, then $A^c \in \mathcal{F}$ (the complement of the set $A$ is also in the field)
2. if $A \in \mathcal{F}$, $B \in \mathcal{F}$, then $A \cup B \in \mathcal{F}$

other properties follow

• $S \in \mathcal{F}$ (where $S$ is the universal set)
• $\emptyset \in \mathcal{F}$ (where $\emptyset$ is the empty set)
• $A \cap B \in \mathcal{F}$
• etc.

Notes

• if $S$ is any collection of subsets of $X$, then we can always find a σ-algebra containing $S$, namely the power set of $S$
• take the intersection of all σ-algebras containing $S$ to obtain the smallest such σ-algebras, the σ-algebra generated by S
• known as a σ-field or simply field in some literature (but not to be confused with the field from field theory/algebra or vector field... terminology is ambiguous)

See also

• topology
• Borel σ-algebra
• Dynkin system

References

1. https://ncatlab.org/nlab/show/sigma-algebra
2. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 239, 344.
3. https://en.wikipedia.org/wiki/Σ-algebra
4. https://math.stackexchange.com/questions/1330649/difference-between-topology-and-sigma-algebra-axioms
5. https://math.stackexchange.com/questions/1080473/borel-sigma-algebra-definition
6. https://www.bananaspace.org/wiki/Sigma-代数