The integral of a nonnegative simple function

Depends on

• Nonnegative simple measurable functions
• Measures on sigma-algebras

Used by

• Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
• The expectation of an indicator is the probability of the event Corollary
• A radial Poisson limit does not control a tangential path Counterexample
• A random variable need not have a finite expectation Counterexample
• A step has no locally integrable weak derivative Counterexample
• Cantor function has singular distributional derivative Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• Strong fractional integration fails at p equal to one Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• Eigenfunction for a probability system Definition
• Rademacher functions on the unit interval Definition
• The nonnegative Lebesgue integral Definition
• A character is positive definite Example
• A square-integrable separable product kernel Example
• Compact groups have a constant Reiter net Example
• Conjugate phases norm a three-atom function Example
• Dilation determines the Riesz-potential target exponent Example
• Dunford--Pettis: dominated and concentrating families Example
• Hilbert transform of the line Poisson kernel Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Mollification of a complex two-step function Example
• Negative drift gives a finite mean small-set hit Example
• Sharp Sobolev threshold for a radial power Example
• The absolute value has a weak first derivative Example
• The Haar orthonormal basis of L²((0,1)) 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
• A bounded Riemann integrable function admits Borel Darboux envelopes with the same Lebesgue integral Lemma
• A topological invariant mean yields norm-approximately invariant densities Lemma
• Averaging a Hermitian form unitarizes a finite-dimensional compact-group representation Lemma
• Borel Darboux integrands in finite dimension Lemma
• Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums Lemma
• Finite Rademacher blocks are equidistributed Lemma
• Følner nets give Reiter nets Lemma
• Fourier-Stieltjes transforms of positive measures are continuous positive definite Lemma
• L1 of a second-countable locally compact group is separable Lemma
• Simultaneous rational conditional distribution function versions Lemma

…and 21 more results.

Dependency tree · two levels

6 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 4.1 (standard reference, not scraped)