The essential supremum of a measurable function with respect to a measure

Depends on

• Measure-null sets and almost-everywhere statements relative to a measure

Used by

• A positive reaction term restores coercivity without Poincar'e Corollary
• The L^∞ maximum bound for entropy solutions Corollary
• The global Harnack comparison needs connectedness Counterexample
• The Harnack estimate needs an additive forcing term Counterexample
• The Harnack inequality requires nonnegativity Counterexample
• The translation semigroup is not strongly continuous on L-infinity Counterexample
• Analytic Hardy spaces on the unit disc Definition
• Complex L^∞ space of a locally compact group Definition
• Complex Lp classes and Euclidean test-function conventions Definition
• Harmonic Hardy classes on the unit disc Definition
• Hᵖ atoms with a prescribed moment order Definition
• Measurable and decomposable operator fields Definition
• Mihlin smoothness convention above half the dimension Definition
• Muckenhoupt Aₚ and A₁ weights Definition
• The Beltrami coefficient and the maximal dilatation Definition
• The space L^∞(μ) of essentially bounded measurable functions Definition
• Uniformly elliptic divergence-form operators and their sesquilinear forms Definition
• A measurable two-dimensional operator field Example
• Compact groups have a constant Reiter net Example
• Indicator of the rationals has zero essential supremum but pointwise supremum one Example
• Pvm of a multiplication operator Example
• The essential supremum precedes the Holder representative in De Giorgi theory Example
• A finite interior ball chain propagates weak Harnack bounds Lemma
• A Reiter net has an invariant-mean cluster point Lemma
• Bounded restriction and cutoff localisation in Sobolev spaces Lemma
• Complex Lq norm recovery from finite simple dual tests Lemma
• De Giorgi oscillation reduction: one half-level set is small Lemma
• Diagonal multipliers form a von Neumann algebra Lemma
• Exact L2 Fourier multiplier norm Lemma
• Moser iteration for positive supersolutions: negative-power and logarithmic comparison Lemma
• Operators commuting with a generating family of multiplications Lemma
• Probability-density approximation of continuous tests and topological means Lemma
• Probability-density averages and locally detectable upper essential values Lemma
• Sobolev functions paste across an overlap Lemma
• The elliptic form is well defined and bounded on H¹ Lemma
• The two defining forms of A₁ agree Lemma
• Zero-set propagation for a nonnegative Holder weak solution Lemma
• Ergodic averages are measurable, representative independent, and Lp contractive Proposition
• The essential supremum is attained as the least essential bound Proposition
• Bounded borel pvm integral Theorem

…and 12 more results.

Dependency tree · two levels

2 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, Definition 7.3 (standard reference, not scraped)
• Sheldon Axler, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)