Difference between revisions of "Sigma-algebra"

Revision as of 12:11, 20 July 2015

A Sigma-algebra of sets, or $\sigma$-algebra is very similar to a $\sigma$-ring of sets.

Like how ring of sets and algebra of sets differ, the same applies to $\sigma$-ring compared to $\sigma$-algebra

Definition

A non empty class of sets $S$ is a $\sigma$-algebra^{[Note 1]} if^[1][2]

• if $$A\in S$$ then $$A^c\in S$$
• if $$\{A_n\}_{n=1}^\infty\subset S$$ then $$\cup^\infty_{n=1}A_n\in S$$

That is it is closed under complement and countable union.

Immediate consequences

Among other things immediately we see that:

Important theorems

Common $\sigma$-algebras

See also: Index of common $\sigma$-algebras

• $\sigma$-algebra generated by
• Trace $\sigma$-algebra
• Pre-image $\sigma$-algebra

See also

• Types of set algebras
• $\sigma$-algebra generated by
• $\sigma$-ring

Notes

1. Jump up ↑ Some books (notably Measures, Integrals and Martingales) give $X\in\mathcal{A}$ as a defining property of $\sigma$-algebras, however the two listed are sufficient to show this (see the immediate consequences section)
2. Jump up ↑ Measures, Integrals and Martingales puts this in the definition of $\sigma$-algebras

References

1. Jump up ↑ Halmos - Measure Theory - page 28 - Springer - Graduate Texts in Mathematics - 18
2. Jump up ↑ Measures, Integrals and Martingales - Rene L. Schilling