Steinhaus theorem
E394469
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Steinhaus property of Lebesgue measure | 1 |
| Steinhaus theorem canonical | 1 |
| Steinhaus theorem for locally compact abelian groups | 1 |
| Steinhaus–Weil theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3884668 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Steinhaus theorem Context triple: [Hugo Steinhaus, notableWork, Steinhaus theorem]
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A.
Ulam problem in set theory
The Ulam problem in set theory is a well-known question posed by Stanislaw Ulam concerning the structure and properties of measurable sets and functions, particularly in relation to homomorphisms and measure-theoretic regularity.
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B.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
C.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
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D.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
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E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Steinhaus theorem Target entity description: The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
-
A.
Ulam problem in set theory
The Ulam problem in set theory is a well-known question posed by Stanislaw Ulam concerning the structure and properties of measurable sets and functions, particularly in relation to homomorphisms and measure-theoretic regularity.
-
B.
Alexandrov–Hausdorff theorem
The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
-
C.
Szekeres–Lindström theorem
The Szekeres–Lindström theorem is a result in combinatorics that characterizes the maximum size of intersecting families of subsets, serving as a precursor to and special case of the Erdős–Ko–Rado theorem.
-
D.
Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
-
E.
Khinchin–Kolmogorov theorem
The Khinchin–Kolmogorov theorem is a fundamental result in probability theory that provides conditions under which series of independent random variables converge almost surely.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
result in measure theory
ⓘ
theorem ⓘ |
| alsoKnownAs |
Steinhaus property of Lebesgue measure
ⓘ
linked to:
Steinhaus theorem
|
| appliesTo | Lebesgue measurable subsets of R with positive measure ⓘ |
| characterizes | non-discrete locally compact groups with respect to Haar measure ⓘ |
| conclusion | 0 is an interior point of the difference set A − A ⓘ |
| doesNotApplyTo | sets of Lebesgue measure zero ⓘ |
| domain | real numbers ⓘ |
| ensures | existence of nontrivial open subset in A − A ⓘ |
| field |
measure theory
ⓘ
real analysis ⓘ |
| generalizationOf | results about density points of measurable sets ⓘ |
| hasGeneralization |
Steinhaus theorem for locally compact abelian groups
ⓘ
linked to:
Steinhaus theorem
Steinhaus–Weil theorem ⓘ
linked to:
Steinhaus theorem
|
| holdsIn | Euclidean spaces R^n with Lebesgue measure ⓘ |
| implies |
sets of positive Lebesgue measure are thick in the sense of containing many differences
ⓘ
the difference set of a set of positive Lebesgue measure is not meagre near 0 ⓘ |
| involvesConcept |
Lebesgue measure
ⓘ
difference set ⓘ open interval ⓘ sets of positive measure ⓘ |
| mathematicalSubjectClassification |
26A30
ⓘ
28A05 ⓘ |
| namedAfter | Hugo Steinhaus ⓘ |
| originalContext | real line with Lebesgue measure ⓘ |
| relatedTo |
Baire category theorem
ⓘ
Lebesgue density theorem ⓘ
linked to:
Lebesgue differentiation theorem
Ruzsa triangle inequality ⓘ sumset phenomena in additive number theory ⓘ |
| requires | positivity of measure ⓘ |
| statement | If A is a subset of the real numbers with positive Lebesgue measure, then the difference set A − A contains an open interval around 0. ⓘ |
| symbolicForm | If m(A) > 0 then ∃ε > 0 such that (−ε, ε) ⊂ A − A. ⓘ |
| typeOfResult | regularity theorem for measurable sets ⓘ |
| usedIn |
additive combinatorics
ⓘ
ergodic theory ⓘ geometric measure theory ⓘ harmonic analysis ⓘ probability theory on the real line ⓘ |
| yearProved | 1920s ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Steinhaus theorem Description of subject: The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Steinhaus theorem
linked to: Steinhaus theorem
linked to: Steinhaus theorem