DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-29

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The polar surface set function on the unit sphere

Definition

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $n \geq 1$ and write $S^{n - 1} : = {ω \in R^{n} : ∣ ω ∣ = 1} .$ Let $Φ : R^{n} ∖ {0} ⟶ (0, \infty) \times S^{n - 1}, Φ (x) = (∣ x ∣, x / ∣ x ∣) .$ This map is continuous. If $E \subseteq S^{n - 1}$ is Borel, then $(0, 1] \times E$ is Borel and ${r ω : ω \in E, 0 < r \leq 1} = Φ^{- 1} ((0, 1] \times E)$ is Borel in $R^{n} ∖ {0}$ by A continuous map has Borel preimages of Borel sets, hence Borel in $R^{n}$ by The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra. It is therefore Lebesgue measurable by Assuming countable choice, every Borel subset of $R^{n}$ is Lebesgue measurable. Define $σ (E) : = n λ_{n} ({r ω : ω \in E, 0 < r \leq 1}) .$

The next theorem proves that this is a Borel measure on $S^{n - 1}$ and that it is exactly the surface measure needed for polar coordinates.

Depends on

Used by

Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr d sigma Theorem

Dependency tree · two levels

26 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

Gerald B. Folland, Real Analysis, 2nd ed., Theorem 2.49 (standard reference, not scraped)