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Integral over a measurable subset
Definition
Let be a measure space, let be measurable, and let . Since Closure properties of measurable functions used by the integral implies that is measurable, define the integral of over by where the integral on the right is the nonnegative Lebesgue integral of The nonnegative Lebesgue integral.
Depends on
Used by
- A nonnegative integral over a null set vanishes Corollary
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere Theorem
- Absolute continuity of the integral Theorem
- Chebyshev-Markov inequality for the integral Theorem
- Monotone convergence for the integral Theorem
- The indefinite integral of a nonnegative measurable function is a measure Theorem
- The indefinite integral of a nonnegative simple function is a measure Theorem
Dependency tree · two levels
5 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)