definition of locally almost-everywhere over measure space
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 definition of locally almost-everywhere over measure space.
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:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\(*S\): \(\subseteq M\)
//
Conditions:
\(M \setminus S \in \{\text{ the locally negligible subsets of } M\}\)
//
Our notation for being locally almost-everywhere is \(l.a.e\), or \(\mu-l.a.e\) if \(\mu\) is needed to be specified.
2: Note
These are some typical usages.
For some maps, \(f_1, f_2: M \to S'\), \(f_1 =_{l.a.e} f_2\), which means that \(\{s \in M \vert f_1 (s) \neq g_2 (s)\}\) is a locally negligible subset of \(M\).
Likewise, \(f_1 \lt_{l.a.e} f_2\) or \(f_1 \le_{l.a.e} f_2\).
