Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-17
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Closed left rays form a pi-system generating the Borel sigma-algebra on the real line

Example

The family P:={(−∞,a]:a∈R} is a pi-system and σR(P)=B(R).

Facts & Assumptions

Given: The family P of closed left rays.

[L1]

A pi-system is a nonempty family closed under binary intersections (Pi-systems).

[L2]

The open right rays (a,∞) generate B(R) (Seven generating families for the Borel sigma-algebra on the real line).

Verification

technique · direct
1.1L1algebra

The family is nonempty, and (−∞,a]∩(−∞,b]=(−∞,min⁡{a,b}], so it is a pi-system by [L1].

2.1step 1.1L2algebra∎

Complementation exchanges (−∞,a] with (a,∞). A generated sigma-algebra is complement-closed, so [L2] implies that the closed left rays generate B(R).

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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