DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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The Borel sigma-algebra on the extended real line

Definition

The extended real line is

$\overline{R} : = R \cup {- \infty, + \infty}$

with the total order extending the usual order on $R$ and satisfying $- \infty < x < + \infty$ for every real $x$ (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined).

Its Borel sigma-algebra is the Borel sigma-algebra of that ordered topological line in the sense of The Borel sigma-algebra of a topological space. On this page it is denoted $B (\overline{R})$ and is generated by the rays

${(a, + \infty] : a \in R} .$

In particular, ${+ \infty}$ and ${- \infty}$ are Borel sets.

Depends on

Used by

Threshold characterisations of real-valued and extended-real-valued measurability Theorem

Dependency tree · two levels

11 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

Sheldon Axler, Measure, Integration and Real Analysis, Section 2B (standard reference, not scraped)