TheoremStatement: AI-adaptedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , let $S^{n - 1} \subseteq R^{n}$ be the unit sphere, and let $σ$ be the set function of The polar surface set function on the unit sphere. Then $σ$ is a finite Borel measure on $S^{n - 1}$ , and for every Borel measurable $f : R^{n} \to [0, \infty]$ , $\int_{R^{n}} f (x) d λ_{n} (x) = \int_{0}^{\infty} \int_{S^{n - 1}} f (r ω) r^{n - 1} d σ (ω) d r .$ Moreover, $σ$ is the unique Borel measure on $S^{n - 1}$ with this property.

Facts & Assumptions

Given: The Axiom of Countable Choice, a positive integer $n$ , and a Borel measurable function $f : R^{n} \to [0, \infty]$ .

[L1]

The set function $σ$ is defined by $σ (E) = n λ_{n} ({r ω : ω \in E, 0 < r \leq 1})$ for Borel $E \subseteq S^{n - 1}$ . (The polar surface set function on the unit sphere)

[L2]

Linear dilations scale Lebesgue measure by the determinant. (A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not)

[L3]

Two measures that agree on a sigma-finite generating pi-system agree on the generated sigma-algebra. (Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system)

[L4]

Tonelli's theorem turns equality of measures on sets into the corresponding equality of nonnegative integrals. (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product)

[L5]

Assuming countable choice, bounded subsets of Euclidean space have finite outer measure. (Lebesgue measure is sigma-finite, and every metrically bounded subset of $R^{n}$ has finite outer measure)

[A1]

Let $Φ : R^{n} ∖ {0} \to (0, \infty) \times S^{n - 1}$ be $Φ (x) = (∣ x ∣, x / ∣ x ∣)$ . This map is continuous, so $m^{*} (B) : = λ_{n} (Φ^{- 1} (B))$ defines a measure on the Borel subsets of $(0, \infty) \times S^{n - 1}$ .

[A2]

Sets of the form $(a, b] \times E$ with $0 < a < b$ and Borel $E \subseteq S^{n - 1}$ form a sigma-finite generating pi-system for the Borel sigma-algebra of $(0, \infty) \times S^{n - 1}$ .

Proof

technique · direct

1.1L1L5A1

For Borel $E \subseteq S^{n - 1}$ put $E_{r} : = {s ω : ω \in E, 0 < s \leq r}$ . By [A1], each $E_{r}$ is Borel. If $(E_{j})$ is pairwise disjoint, then the sets $(E_{j})_{1}$ are pairwise disjoint and $(⋃_{j} E_{j})_{1} = ⋃_{j} (E_{j})_{1}$ , so countable additivity of $λ_{n}$ makes $σ$ a Borel measure. It is finite because $σ (S^{n - 1}) = n λ_{n} ({x : 0 < ∣ x ∣ \leq 1}) < \infty$ by [L5].

2.1L1L2A1step 1.1

For Borel $E \subseteq S^{n - 1}$ and $0 < a < b$ , [L1] gives $λ_{n} (E_{1}) = σ (E) / n$ . The dilation $x \mapsto r x$ has determinant $r^{n}$ , so [L2] gives $λ_{n} (E_{r}) = r^{n} λ_{n} (E_{1}) = \frac{r^{n}}{n} σ (E) .$ Therefore $m^{*} ((a, b] \times E) = λ_{n} (E_{b} ∖ E_{a}) = \frac{b^{n} - a^{n}}{n} σ (E) = σ (E) \int_{a}^{b} r^{n - 1} d r .$

3.1A2L3L4step 1.1step 2.1∎

Step 2.1 shows that $m^{*}$ and the product measure $r^{n - 1} d r \times σ$ agree on the generating pi-system of [A2]. The rectangles $P_{k} = (1 / k, k] \times S^{n - 1}$ for $k \geq 2$ increase to the whole product and have the same finite measure under both measures, namely $σ (S^{n - 1}) (k^{n} - k^{- n}) / n$ ; hence [L3] applies and the measures agree on every Borel subset of $(0, \infty) \times S^{n - 1}$ . Applying [L4] to this identity gives the integral formula on $R^{n} ∖ {0}$ . The omitted origin has Lebesgue measure zero: it lies in a half-open box of volume $ε^{n}$ for every $ε > 0$ . Thus the same formula holds for every nonnegative Borel measurable $f$ on all of $R^{n}$ , even when $f (0) = + \infty$ . If $\tilde{σ}$ is another Borel measure on $S^{n - 1}$ with the same integral formula, apply that formula to $1_{E_{1}}$ , where $E_{1} = {r ω : ω \in E, 0 < r \leq 1}$ . Then $λ_{n} (E_{1}) = \int_{0}^{\infty} \int_{S^{n - 1}} 1_{E_{1}} (r ω) r^{n - 1} d \tilde{σ} (ω) d r = \tilde{σ} (E) \int_{0}^{1} r^{n - 1} d r = \frac{\tilde{σ} (E)}{n},$ so $\tilde{σ} (E) = n λ_{n} (E_{1}) = σ (E)$ by [L1]. Thus $\tilde{σ} = σ$ , proving uniqueness.

Depends on

Used by

Ball mean-value property for harmonic functions under Countable Choice Corollary
Interior gradient bound for Poisson solutions Corollary
Polar integration may discard the cut locus Corollary
The Hilbert transform is bounded on Lp Corollary
The Riesz transforms are bounded on Lp Corollary
A non-Dini continuous Poisson source can destroy the continuity of the second derivatives Counterexample
A power weight fails at both Aₚ endpoints Counterexample
Ambient-smooth density fails on a slit disc Counterexample
An inward cusp blocks W^1,3/2 extension Counterexample
Boundary H² regularity needs domain regularity Counterexample
Boundary point values are not a function of the interior Lᵖ class Counterexample
Boundary W^2,p regularity needs more than Lipschitz boundary Counterexample
Interior regularity does not imply boundary regularity Counterexample
Mixing scaled and unscaled Minkowski covolumes fails Counterexample
Morrey's inequality has no p=n endpoint Counterexample
One centred ball-mean identity does not force harmonicity Counterexample
Outward dilation defeats subcritical inclusion and homogeneous Poincare on Euclidean space Counterexample
Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
Size without cancellation does not give a principal value Counterexample
Strong fractional integration fails at p equal to one Counterexample
Subcritical W^1,p is not closed under multiplication Counterexample
The complementarity product needs extra regularity Counterexample
The critical Riesz potential can diverge and be essentially unbounded Counterexample
The fundamental Hessian is not absolutely locally integrable Counterexample
Fundamental solution for the positive operator minus Laplacian Definition
Maximal truncated singular integrals Definition
Model space radial area and ball volume Definition
Spherical averages and local ball means in Rn Definition
Spherical means and the weighted ball integral of space-dependent data Definition
A compact-support Cauchy–Pompeiu calculation Example
A Dirac mass has precisely sufficiently negative Sobolev order Example
An explicit ∂̄ solution with an L² estimate Example
Dilation determines the Riesz-potential target exponent Example
Hörmander estimate with a Gaussian weight Example
Newtonian Hessian kernels fit the Calderón–Zygmund framework Example
Newtonian potential of radial compact data Example
Projective-plane curvature via a hemisphere Example
Radial powers approach the Morrey borderline exponent Example
Sharp Sobolev threshold for a radial power Example
The A₁ range of a power weight Example

…and 53 more results.

Dependency tree · two levels

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Sources

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