Integral over a measurable subset

Depends on

• The nonnegative Lebesgue integral
• Closure properties of measurable functions used by the integral

Used by

• A nonnegative integral over a null set vanishes Corollary
• Dominated convergence is a Vitali corollary Corollary
• Polar integration may discard the cut locus Corollary
• Positive, negative, and truncated Sobolev functions Corollary
• Weak differentiation has a closed graph on its natural domains Corollary
• A large adverse zero-order term destroys Dirichlet coercivity Counterexample
• A radial Poisson limit does not control a tangential path Counterexample
• A step has no locally integrable weak derivative Counterexample
• Ambient-smooth density fails on a slit disc Counterexample
• Neumann Poisson data require a flux compatibility equation Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Strong fractional integration fails at p equal to one Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• The Neumann Poisson problem is not coercive on all of H¹ Counterexample
• A tight family of integrable functions Definition
• A uniformly integrable family Definition
• Indefinite Lebesgue integral on a compact interval Definition
• The circle maximal function and nontangential approach regions Definition
• The Lusin area function for a fixed admissible kernel and aperture Definition
• A disconnected Neumann domain has a multiple zero eigenvalue Example
• Dilation determines the Riesz-potential target exponent Example
• Hilbert transform of the line Poisson kernel Example
• L² forcing defines an H⁻¹ functional Example
• Matching C¹ pieces across a hyperplane have no jump derivative Example
• Newtonian potential of radial compact data Example
• Sharp Sobolev threshold for a radial power Example
• The absolute value has a weak first derivative Example
• The Neumann kernel is spanned by the componentwise constants Example
• The square of the Volterra operator has zero trace Example
• The weak Dirichlet Poisson problem on an interval Example
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• Bounded restriction and cutoff localisation in Sobolev spaces Lemma
• Coercivity of the principal Dirichlet form Lemma
• Compactly supported Sobolev functions extend by zero in every integer order Lemma
• Finite Rademacher blocks are equidistributed Lemma
• K-type eigenvalues of A(nu): recurrence, closed form and nonvanishing Lemma
• Simultaneous rational conditional distribution function versions Lemma
• Smooth compactly supported functions of an open set are dense in L² Lemma
• Sobolev functions paste across an overlap Lemma

…and 24 more results.

Dependency tree · two levels

8 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)