definition of function Lebesgue integrable 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 function Lebesgue integrable over measurable subset of measure space.
Orientation
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Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( (M, A, \mu)\): \(\in \{\text{ the measure spaces }\}\)
\( \overline{\mathbb{R}}\): \(= \text{ the extended Euclidean topological space }\) with the Borel \(\sigma\)-algebra
\(*f\): \(: M \to \overline{\mathbb{R}}\), \(\in \{\text{ the measurable maps }\}\)
\( a\): \(\in A\)
\( \chi_a\): \(= \text{ the characteristic function of } a\)
\( f^+\): \(: M \to [0, \infty], s \mapsto max (0, f (s))\)
\( f^-\): \(: M \to [0, \infty], s \mapsto - min (0, f (s))\)
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Conditions:
\(\int_M \chi_a f^+ d \mu \lt \infty \land \int_M \chi_a f^- d \mu \lt \infty\)
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When \(a = M\), \(f\) is called "Lebesgue integrable function".
