definition of complex function Lebesgue integrable over measurable subset of measure space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of complex Euclidean topological space.
- The reader knows a definition of Borel \(\sigma\)-algebra of topological space.
- The reader knows a definition of measurable map between measurable spaces.
- The reader knows a definition of extended real function Lebesgue integrable over measurable subset of measure space.
Target Context
- The reader will have a definition of complex function Lebesgue integrable over measurable subset of 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:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\( \mathbb{C}\): \(= \text{ the complex Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\(*f\): \(: M \to \mathbb{C}\), \(\in \{\text{ the measurable maps }\}\)
\( a\): \(\in A\)
\( \chi_a\): \(= \text{ the characteristic function of } a\)
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Conditions:
\(Re (f) \in \{\text{ the extended real functions Lebesgue integrable over } a\} \land Im (f) \in \{\text{ the extended real functions Lebesgue integrable over } a\}\)
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When \(a = M\), \(f\) is called "Lebesgue integrable function".
