Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Measure-null sets and almost-everywhere statements relative to a measure
Definition
In a measure space (Measure spaces), a measurable set is -null if .
A property holds -almost everywhere, or for -almost every , if its exceptional set is contained in a measurable -null set: there is with such that holds for every . Both notions are relative to the named measure .
Depends on
Used by
- A bounded function on a closed bounded interval, or on a closed nondegenerate rectangle, is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero Corollary
- A nonnegative integral over a null set vanishes Corollary
- A property holding outside a set of elementary measure zero is exactly a property holding λ-almost everywhere Corollary
- Polar integration may discard the cut locus Corollary
- Equality almost surely is not pointwise equality Counterexample
- A metric for convergence in probability Definition
- Add, cov, non and cof for null and meagre ideals Definition
- Complete measure spaces Definition
- Convergence almost everywhere relative to a measure Definition
- Direct integral of a measurable Hilbert field Definition
- Strict and mod-null invariant sigma-algebras Definition
- Strongly measurable Banach-valued function Definition
- The completion domain and proposed completed set function of a measure space Definition
- The essential supremum of a measurable function with respect to a measure Definition
- The null subspace of measurable functions that vanish almost everywhere Definition
- The Radon-Nikodym derivative as an almost-everywhere equivalence class Definition
- Bounded compact data give an everywhere finite Newtonian potential Lemma
- Elementary bounds on ideal cardinal invariants Lemma
- A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ℝⁿ Proposition
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0 Proposition
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere Theorem
- Almost every point is a Lebesgue point of a locally integrable function Theorem
- Convergence in measure determines the limit almost everywhere Theorem
- Newtonian potentials solve the distributional Poisson equation Theorem
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
- G. Folland, Real Analysis, 2nd ed., §1.3 (standard reference, not scraped)