description/proof of that for set and \(2\) sets of subsets, if former set of subsets is contained in latter set of subsets, \(\sigma\)-algebra generated by former set is contained in \(\sigma\)-algebra generated by latter set
Topics
About: measurable space
The table of contents of this article
Starting Context
- The reader knows a definition of \(\sigma\)-algebra of set generated by set of subsets.
Target Context
- The reader will have a description and a proof of the proposition that for any set and any \(2\) sets of subsets, if the former set of subsets is contained in the latter set of subsets, the \(\sigma\)-algebra generated by the former set is contained in the \(\sigma\)-algebra generated by the latter set.
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:
\(S'\): \(\in \{\text{ the sets }\}\)
\(S_1\): \(\subseteq Pow (S')\)
\(S_2\): \(\subseteq Pow (S')\)
//
Statements:
\(S_1 \subseteq S_2\)
\(\implies\)
\(\sigma (S_1) \subseteq \sigma (S_2)\)
//
2: Proof
Whole Strategy: Step 1: see that \(S_1 \subseteq \sigma (S_2)\) and \(\sigma (S_1) \subseteq \sigma (S_2)\).
Step 1:
\(S_1 \subseteq S_2 \subseteq \sigma (S_2)\), by the definition of \(\sigma\)-algebra of set generated by set of subsets.
So, \(\sigma (S_2)\) is a \(\sigma\)-algebra such that \(S_1 \subseteq \sigma (S_2)\), so, \(\sigma (S_1) \subseteq \sigma (S_2)\), by the definition of \(\sigma\)-algebra of set generated by set of subsets.
