The nonnegative Lebesgue integral

Depends on

• Extended-real-valued measurable functions
• Nonnegative simple measurable functions
• The integral of a nonnegative simple function

Used by

• A nonnegative integral over a null set vanishes Corollary
• Flat hyperplanes do not have spherical stationary-phase decay Counterexample
• Knapp rules out extension below the Tomas exponent Counterexample
• Countable partition construction of the Borel set function Definition
• Direct integral of a measurable Hilbert field Definition
• Eigenfunction for a probability system Definition
• Expectation of a nonnegative or integrable random variable Definition
• Extremal length and the curve-family modulus of a path family Definition
• Integrable real and complex functions, and their integrals Definition
• Integral over a measurable subset Definition
• Surface integration on compact C1 hypersurfaces Definition
• The ACL and Sobolev analytic definition of quasiconformality Definition
• The function space Lᵖ(μ) for 0 < p < ∞ Definition
• The Gagliardo--Slobodeckij space on Euclidean space Definition
• Wave energy density, energy flux and total energy Definition
• A character is positive definite Example
• A square-integrable separable product kernel Example
• Compact groups have a constant Reiter net Example
• Counting measure represents finite-support summation on a discrete LCH space Example
• Finite counting measure on n points recovers ℝⁿ p-norms Example
• Integrating against a Dirac measure is evaluation at the point Example
• Integrating against counting measure recovers a series Example
• Negative drift gives a finite mean small-set hit Example
• Point evaluation is represented by a Dirac measure Example
• The regular representation of the real line as a multiplicity-one integral of characters Example
• The square of the Volterra operator has zero trace Example
• Two-step functions expose the L² conjugation convention Example
• Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
• Decay of a localized measure on a curved graph patch Lemma
• Fourier-Stieltjes transforms of positive measures are continuous positive definite Lemma
• Probability-density averages and locally detectable upper essential values Lemma
• Simultaneous rational conditional distribution function versions Lemma
• Stationary phase with a compactly supported amplitude Lemma
• The Hardy inequality for the averaging operator on the half-line Lemma
• The rho-length and the extremal length are well defined Lemma
• The unit sphere is Lebesgue null Lemma
• Van der Corput oscillatory integral estimates in one dimension Lemma
• Compact and locally compact abelian groups are amenable Proposition
• Monotonicity and nonnegative homogeneity of the nonnegative integral Proposition
• The nonnegative integral agrees with the simple integral on simple functions Proposition

…and 15 more results.

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)