Alphabeta Math

Sigma Algebras and Borel Sets — Examples

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How 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.

✓ 14 results · all verified · 12 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Measure Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

The trivial and discrete sigma-algebras are the two extremes

Example

Assuming countable choice, the countable-cocountable family is a sigma-algebra

Example

A partition into k nonempty blocks generates a sigma-algebra with 2ᵏ members

Example

F-sigma and G-delta subsets of the real line are Borel

Example

The rationals are Borel and F-sigma but neither open nor closed nor G-delta

Example

Closed left rays form a pi-system generating the Borel sigma-algebra on the real line

Example

The Borel sigma-algebra of the Cantor set is the trace of the real Borel sigma-algebra

False statement

FALSE: every lambda-system is closed under finite intersections

False statement

FALSE: the union of an increasing sequence of monotone classes is a monotone class

False statement

FALSE: the union of an increasing sequence of sigma-algebras is a sigma-algebra

False statement

FALSE: the union of two sigma-algebras on one set is a sigma-algebra

False statement

FALSE: every monotone class is an algebra

False statement

FALSE: every subset of the real line is Borel

False statement

FALSE: a countably infinite sigma-algebra exists

False statement

FALSE: every Borel subset of the real line is a countable union of countable intersections of open and closed sets

Sources