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.
The Dirichlet function is positive on a dense set but has Lebesgue integral
Statement refuted
A nonnegative function that is positive on a dense subset of must have strictly positive Lebesgue integral.
Facts & Assumptions
Given: The Dirichlet function on .
The Dirichlet function is the indicator of the rational reals (The Dirichlet function , and Thomae's function with at a rational in lowest terms with and at every irrational ).
The rationals in are Lebesgue null (Every at most countable subset of is Lebesgue null; in particular ).
A nonnegative measurable function has integral exactly when it vanishes almost everywhere (A nonnegative measurable function has integral exactly when it vanishes almost everywhere).
Counterexample
By [L1], the restriction of to is positive at every rational point, hence on a dense subset of .
By [L2], the set on which is positive is null. Therefore almost everywhere on , so [L3] gives.
This refutes the Statement.
Depends on
- The Dirichlet function $1_{\mathbb{Q}}$, and Thomae's function $t$ with $t(x) = 1/q$ at a rational $x = p/q$ in lowest terms with $q \ge 1$ and $t(x) = 0$ at every irrational $x$
- Every at most countable subset of $\mathbb{R}^n$ is Lebesgue null; in particular $\lambda_1(\mathbb{Q})=0$
- A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
29 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, Example 4.2 (standard reference, not scraped)