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.
Equality almost surely is not pointwise equality
Statement refuted
If two random variables are equal almost surely, then they are pointwise equal.
Facts & Assumptions
Given: The probability space , the zero function , and the indicator .
A property holds almost everywhere when its exceptional set is contained in a measurable null set (Measure-null sets and almost-everywhere statements relative to a measure).
Integrable random variables with the same almost-sure class have the same expectation (Expectation depends only on the almost-everywhere class).
Counterexample
The functions and differ only at the single point . That set is Lebesgue-null, so [L1] gives almost surely.
They are not pointwise equal, because while . They are both bounded and hence integrable, so [L2] also gives
Therefore almost-sure equality is strictly weaker than pointwise equality.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
5 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.5 (standard reference, not scraped)