definition of almost-everywhere over measure space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of negligible subset of measure space.
Target Context
- The reader will have a definition of 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:
\(M \setminus S \in \{\text{ the negligible subsets of } M\}\)
//
A prevalent notation for being almost-everywhere is \(a.e\), or \(\mu-a.e\) if \(\mu\) is needed to be specified.
2: Note
These are some typical usages.
For some maps, \(f_1, f_2: M \to F\), \(f =_{a.e} g\), which means that \(\{m \in M \vert f (m) \neq g (m)\}\) is a negligible subset of \(M\).
Likewise, \(f \lt_{a.e} g\) or \(f \le_{a.e} g\).
