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The nonnegative Lebesgue integral
Definition
Let be measurable (Extended-real-valued measurable functions). Its nonnegative Lebesgue integral is where the simple integral on the right is the one from The integral of a nonnegative simple function.
The set of admissible simple minorants is nonempty because it contains the zero function, which is simple (Nonnegative simple measurable functions).
Depends on
Used by
- A nonnegative integral over a null set vanishes Corollary
- Integrable real and complex functions, and their integrals Definition
- Integral over a measurable subset Definition
- Integrating against a Dirac measure is evaluation at the point Example
- Integrating against counting measure recovers a series Example
- Monotonicity and nonnegative homogeneity of the nonnegative integral Proposition
- The nonnegative integral agrees with the simple integral on simple functions Proposition
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere Theorem
- Jensen's integral inequality for a probability measure Theorem
- Monotone convergence for the integral Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John K. Hunter, Measure Theory Notes, Definition 4.4 (standard reference, not scraped)
- Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)