View source for Sigma-algebra generated by

See [[Notes:Just what is in a generated sigma-algebra|Just what is in a generated {{Sigma|algebra}}]] for examples
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==Theorem statement==
Given a set {{M|S\subseteq\mathcal{P}(\Omega)}} (where {{M|\mathcal{P}(\Omega)}} denotes the [[Power set|power set]] of {{M|\Omega}}) there exists<ref name="PTACC">Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke</ref> a smallest [[Sigma-algebra|{{Sigma|algebra}}]] which we denote {{M|\sigma(S)}} such that:
* {{M|S\subseteq\sigma(S)}} where <mm>\sigma(S)=\bigcap_{\mathcal{A}\subseteq\mathcal{P}(\Omega)\text{ is a }\sigma\text{-algebra}\wedge S\subseteq\mathcal{A}}\mathcal{A}</mm>
We say:
* {{M|\sigma(S)}} the {{Sigma|algebra}} generated by {{M|S}}
* {{M|S}} the generator of {{M|\sigma(S)}}
{{Begin Theorem}}
Proof:
{{Begin Proof}}
We will prove:
# First that there actually is a {{Sigma|algebra}} that contains {{M|S}}
# Then there is a smallest {{Sigma|algebra}} that contains {{M|S}}


'''Proof:'''
# As {{M|\mathcal{P}(\Omega)}} is a {{Sigma|algebra}} we know a sigma algebra containing {{M|S}} exists.
#* So the intersection in the definition is non empty.
#* By [[The intersection of sets is a subset of each set]] we now see that {{M|\sigma(S)\subseteq\mathcal{P}(\Omega)}}
# Using [[Sigma-algebra#First theorems|the intersection of {{Sigma|algebras}} is a {{Sigma|algebra}}]] (for an arbitrary [[Indexing set|indexing set]])
#* If the indexing set is all the {{Sigma|algebras}} that contain {{M|S}} then we see immediately that {{M|\sigma(S)}} is the smallest {{Sigma|algebra}} containing {{M|S}}

The statement is proved
{{End Proof}}{{End Theorem}}
==See also==
* [[Ring generated by]]
* [[Borel sigma-algebra|Borel {{Sigma|algebra}}]]

==References==
<references/>

{{Theorem Of|Measure Theory}}