= σ Algebra =

A '''σ algebra''' (sigma algebra) is a set of measurable sets.

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== Description ==

A σ algebra is usually notated using a calligraphic uppercase letters, like 𝒜.

For a given sample space ''Ω'', a σ algebra 𝒜 can be defined. It is the [[Analysis/Sets|subset]] of the [[Analysis/PowerSet|power set]] of the sample space (𝒜 ''⊆ P(Ω)'') that satisfies three conditions:

 * Both the empty set and the full sample space are in the σ algebra; ''Ø, Ω ∈'' 𝒜
 * The sigma algebra is closed on complementation; ''A ∈'' 𝒜 ''-> A^C^ ∈'' 𝒜
 * The sigma algebra is closed upon [[Analysis/Cardinality|countable]] unions; ''⋃ A,,i,, ∈'' 𝒜 where members of 𝒜 be indexed as ''A,,i,,'', ''i ∈ N''

The smallest possible σ algebra is given by condition 1 literally: ''{Ø, Ω}''. The largest possible σ algebra is of course ''P(Ω)'' itself. All possible σ algebra are bounded by these two extrema.



=== Properties ===

For a given sample space ''Ω'', and some number of σ algebras 𝒜'',,i,,'' defined on it, ''⋂,,i,,'' 𝒜 '',,i,,'' is also a σ algebra.

It follows that for any ℳ ''⊆ P(Ω)'', the smallest possible σ algebra that contains ℳ (or the σ algebra generated by ℳ) is given by the intersection of every 𝒜 larger than ℳ. This operation is notated as:

{{attachment:gen.svg}}

The '''Borel set''' (or Borel σ algebra) defined on ''Ω'' is the smallest possible σ algebra generated by the [[Analysis/TopologicalSpace|topology]] of ''Ω''. It follows that:

{{attachment:borel.svg}}



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