description/proof of that for measure space, union of sequence of locally negligible subsets is locally negligible
Topics
About: measure space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any measure space, the union of any sequence of any locally negligible subsets 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:
\((M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\(s\): \(: \mathbb{N} \to \{\text{ the locally negligible subsets of } M\}\)
//
Statements:
\(\cup_{j \in \mathbb{N}} s (j) \in \{\text{ the locally negligible subsets of } M\}\)
//
2: Proof
Whole Strategy: Step 1: take any \(a \in A\) such that \(\mu (a) \lt \infty\), and see that \((\cup_{j \in \mathbb{N}} s (j)) \cap a\) is negligible.
Step 1:
Let \(a \in A\) be any such that \(\mu (a) \lt \infty\).
Let us see that \((\cup_{j \in \mathbb{N}} s (j)) \cap a\) is negligible.
\((\cup_{j \in \mathbb{N}} s (j)) \cap a = \cup_{j \in \mathbb{N}} (s (j) \cap a)\), by the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset.
As \(s (j)\) is locally negligible, \(s (j) \cap a \subseteq N_j\) for an \(N_j \in A\) such that \(\mu (N_j) = 0\).
So, \((\cup_{j \in \mathbb{N}} s (j)) \cap a \subseteq \cup_{j \in \mathbb{N}} N_j\).
\(\cup_{j \in \mathbb{N}} N_j \in A\), and \(\mu (\cup_{j \in \mathbb{N}} N_j) \le \sum_{j \in \mathbb{N}} \mu (N_j) = \sum_{j \in \mathbb{N}} 0 = 0\).
So, \((\cup_{j \in \mathbb{N}} s (j)) \cap a\) is negligible.
So, \(\cup_{j \in \mathbb{N}} s (j)\) is locally negligible.
