definition of Lebesgue integral of integrable complex function over measurable subset of measure space
Topics
About: measure space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of Lebesgue integral of integrable complex function over measurable subset of measure space.
Orientation
There is a list of definitions discussed so far in this site.
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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
\( a\): \(\in A\)
\( f\): \(: M \to \mathbb{C}\), \(\in \{\text{ the functions integrable over } a\}\)
\( \chi_a\): \(= \text{ the characteristic function of } a\)
\(*\int_a f d \mu\): \(= \int_M \chi_a Re (f) d \mu + i \int_M \chi_a Im (f) d \mu\)
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Conditions:
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2: Note
Lebesgue integral of complex function is defined only for any integrable function, while Lebesgue integral of extended real function is not so, because for example, we have not defined \(\infty + i\) or \(1 + i \infty\).
