definition of \(\sigma\)-algebra of set generated by set of subsets
Topics
About: measure
The table of contents of this article
Starting Context
- The reader knows a definition of \(\sigma\)-algebra of set.
Target Context
- The reader will have a definition of \(\sigma\)-algebra of set generated by set of subsets.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( S'\): \(\in \{\text{ the sets }\}\)
\( S\): \(\subseteq Pow S'\)
\(*\sigma (S)\): \(\in \{\text{ the } \sigma \text{ -algebras of } S'\}\)
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Conditions:
\(S \subseteq \sigma (S)\)
\(\land\)
\(\forall A' \in \{\text{ the } \sigma \text{ -algebras of } S'\} \text{ such that } S \subseteq A' (\sigma (S) \subseteq A')\)
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2: Note
In other words, \(\sigma (S)\) is the smallest \(\sigma\)-algebra that contains \(S\).
\(\sigma (S)\) is uniquely determined, because it is the intersection of all the \(\sigma\)-algebras of \(S'\) that contain \(S\), while at least, \(Pow S'\) is such a one: the intersection is a \(\sigma\)-algebra of \(S'\) that contains \(S\), by the proposition that for any set, the intersection of any \(\sigma\)-algebras is a \(\sigma\)-algebra; for each \(A' \in \{\text{ the } \sigma \text{ -algebras of } S'\} \text{ such that } S \subseteq A'\), \(\sigma (S) \subseteq A'\), because \(A'\) is a constituent of the intersection.
