description/proof of that for measure space, subset of locally negligible subset of space is locally negligible
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of locally negligible subset of measure space.
Target Context
- The reader will have a description and a proof of the proposition that for any measure space, any subset of any locally negligible subset of the space is locally negligible.
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:
\((M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\(S'\): \(\in \{\text{ the locally negligible subsets of } M\}\)
\(S\): \(\subseteq S'\)
//
Statements:
\(S \in \{\text{ the locally negligible subsets of } M\}\)
//
2: Proof
Whole Strategy: Step 1: take any \(a \in A\) such that \(\mu (a) \lt \infty\), and see that \(S \cap a\) is a negligible subset of \(M\).
Step 1:
Let \(a \in A\) be any such that \(\mu (a) \lt \infty\).
\(S \cap a \subseteq S' \cap a\).
But as \(S'\) is locally negligible, there is an \(N \in A\) such that \(S' \cap a \subseteq N\) and \(\mu (N) = 0\).
So, \(S \cap a \subseteq S' \cap a \subseteq N\) and \(\mu (N) = 0\).
So, \(S\) is locally negligible.
