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A radial transverse wave-vector integral has the isotropic factor

Statement

Assume Countable Choice, let $c > 0$ , let $i, j \in {1, 2, 3}$ , and let a Borel function $u : (0, \infty) \to C$ satisfy $\int_{0}^{\infty} ∣ u (ω) ∣ ω^{2} d ω < \infty$ . On $R^{3}$ define $v (k) = u (c ∣ k ∣)$ for $k \neq 0$ and $v (0) = 0$ . Define $Q_{i j} (k) = δ_{i j} - k_{i} k_{j} / ∣ k ∣^{2}$ for $k \neq 0$ and $Q_{i j} (0) = 0$ , where $δ_{i j}$ is $1$ if $i = j$ and $0$ otherwise. All integrals are Lebesgue integrals.

Then $v$ and $v Q_{i j}$ are integrable on $R^{3}$ , and

$\int_{R^{3}} v (k) Q_{i j} (k) d^{3} k = \frac{8 π}{3 c^{3}} δ_{i j} \int_{0}^{\infty} u (ω) ω^{2} d ω .$

The same assertion holds with $u$ replaced by $u 1_{E}$ for any Borel frequency set $E \subseteq (0, \infty)$ , using the same zero extension; this includes every Borel frequency shell.

Facts & Assumptions

Given: Countable Choice, the positive constant $c$ , the indices, Borel function, weighted-integrability hypothesis and zero extensions in the Statement.

[F1]

Under Countable Choice, polar surface measure $σ$ on $S^{2}$ is a finite Borel measure, and the polar formula decomposes each nonnegative Borel integral on $R^{3}$ into $r^{2} d r d σ$ . Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ

[F2]

Fubini applies to absolutely integrable functions on a product of sigma-finite measure spaces. Fubini's theorem for L¹ functions on a sigma-finite product

[F3]

Under Countable Choice, the dilation $r \mapsto c r$ scales one-dimensional Lebesgue measure by $c$ . A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not

[F4]

With $q_{i j} (n) = δ_{i j} - n_{i} n_{j}$ on $S^{2}$ , the angular identity is $\int_{S^{2}} q_{i j} (n) d σ (n) = 8 π δ_{i j} / 3$ . Transverse projector and isotropic angular contraction

[F5]

Countable Choice is assumed for the polar, angular and Lebesgue-measure suppliers. The Axiom of Countable Choice (AC_ω)

[F6]

The integral is complex-linear on integrable functions. The Lebesgue integral is linear on L¹(μ)

[F7]

The nonnegative integral is the supremum of the integrals of nonnegative simple minorants. The nonnegative Lebesgue integral

[F8]

Complex integrability means finite integral of the modulus; the complex integral combines the real and imaginary integrals, each defined through positive and negative parts. Integrable real and complex functions, and their integrals

[F9]

Under Countable Choice, Euclidean Lebesgue measure is sigma-finite and bounded measurable sets have finite measure. Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure

[F10]

Tonelli identifies a nonnegative product integral with its iterated integrals for sigma-finite factor measures. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F11]

The polar surface set function used here is $σ (E) = 3 λ_{3} ({r n : n \in E, 0 < r \leq 1})$ for Borel $E \subseteq S^{2}$ . Its measure property is supplied by [F1]. The polar surface set function on the unit sphere

[F12]

The Borel sigma-algebra is generated by open sets. A map is measurable when preimages of measurable target sets are measurable; compositions of measurable maps are measurable. Identify $C$ with $R^{2}$ with its Borel sigma-algebra. The Borel sigma-algebra of a topological space A measurable function between measurable spaces Composition with a Borel measurable outer map preserves measurability

[F13]

Continuous maps between topological spaces have Borel preimages of Borel sets, and a subspace has the trace of the ambient Borel sigma-algebra. Under Countable Choice, Euclidean Borel sets are Lebesgue measurable. Here $d^{3} k$ and $d r$ mean $λ_{3}$ and the restriction of $λ_{1}$ to $(0, \infty)$ , respectively, with their Lebesgue sigma-algebras. A continuous map has Borel preimages of Borel sets The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ

[F14]

A product sigma-algebra is generated by measurable rectangles; for Euclidean factors it equals the Borel sigma-algebra of the Euclidean product. The product sigma-algebra and its finite iterates The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ

[F15]

For a nonnegative simple function $s = \sum_{ℓ = 1}^{m} a_{ℓ} 1_{E_{ℓ}}$ with disjoint measurable sets, its simple integral is $\sum_{ℓ} a_{ℓ} μ (E_{ℓ})$ , using $0 \cdot \infty = 0$ , and equals its nonnegative Lebesgue integral. In particular $\int 1_{X} d μ = μ (X)$ . The integral of a nonnegative simple function The nonnegative integral agrees with the simple integral on simple functions

[F16]

Nonnegative integrals are monotone and satisfy $\int a f d μ = a \int f d μ$ for every finite $a > 0$ ; the zero function has integral zero. Monotonicity and nonnegative homogeneity of the nonnegative integral

[F17]

A measure is finite when its total mass is finite, and sigma-finite when its space is a countable union of measurable sets of finite measure. In particular a finite measure is sigma-finite, using the whole space at every index. Finite, sigma-finite, and semifinite measures

[F18]

For sigma-finite measure spaces $(X, A, μ)$ and $(Y, B, ν)$ , their product measure on $A \otimes B$ is defined by $(μ \times ν) (E) = \int_{X} ν (E_{x}) d μ = \int_{Y} μ (E^{y}) d ν$ , where $E_{x} = {y : (x, y) \in E}$ and $E^{y} = {x : (x, y) \in E}$ . The product measure of two sigma-finite measure spaces

Proof

Given: Countable Choice, the positive constant $c$ , the indices, Borel function, weighted-integrability hypothesis and zero extensions in the Statement.

[F1] Under Countable Choice, polar surface measure $σ$ on $S^{2}$ is a finite Borel measure, and the polar formula decomposes each nonnegative Borel integral on $R^{3}$ into $r^{2} d r d σ$ . Polar coordinates decompose Lebesgue measure into rⁿ⁻¹ dr dσ

[F2] Fubini applies to absolutely integrable functions on a product of sigma-finite measure spaces. Fubini's theorem for L¹ functions on a sigma-finite product

[F3] Under Countable Choice, the dilation $r \mapsto c r$ scales one-dimensional Lebesgue measure by $c$ . A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not

[F4] With $q_{i j} (n) = δ_{i j} - n_{i} n_{j}$ on $S^{2}$ , the angular identity is $\int_{S^{2}} q_{i j} (n) d σ (n) = 8 π δ_{i j} / 3$ . Transverse projector and isotropic angular contraction

[F5] Countable Choice is assumed for the polar, angular and Lebesgue-measure suppliers. The Axiom of Countable Choice (AC_ω)

[F6] The integral is complex-linear on integrable functions. The Lebesgue integral is linear on L¹(μ)

[F7] The nonnegative integral is the supremum of the integrals of nonnegative simple minorants. The nonnegative Lebesgue integral

[F8] Complex integrability means finite integral of the modulus; the complex integral combines the real and imaginary integrals, each defined through positive and negative parts. Integrable real and complex functions, and their integrals

[F9] Under Countable Choice, Euclidean Lebesgue measure is sigma-finite and bounded measurable sets have finite measure. Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure

[F10] Tonelli identifies a nonnegative product integral with its iterated integrals for sigma-finite factor measures. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product

[F11] The polar surface set function used here is $σ (E) = 3 λ_{3} ({r n : n \in E, 0 < r \leq 1})$ for Borel $E \subseteq S^{2}$ . Its measure property is supplied by [F1]. The polar surface set function on the unit sphere

[F12] The Borel sigma-algebra is generated by open sets. A map is measurable when preimages of measurable target sets are measurable; compositions of measurable maps are measurable. Identify $C$ with $R^{2}$ with its Borel sigma-algebra. The Borel sigma-algebra of a topological space A measurable function between measurable spaces Composition with a Borel measurable outer map preserves measurability

[F13] Continuous maps between topological spaces have Borel preimages of Borel sets, and a subspace has the trace of the ambient Borel sigma-algebra. Under Countable Choice, Euclidean Borel sets are Lebesgue measurable. Here $d^{3} k$ and $d r$ mean $λ_{3}$ and the restriction of $λ_{1}$ to $(0, \infty)$ , respectively, with their Lebesgue sigma-algebras. A continuous map has Borel preimages of Borel sets The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ

[F14] A product sigma-algebra is generated by measurable rectangles; for Euclidean factors it equals the Borel sigma-algebra of the Euclidean product. The product sigma-algebra and its finite iterates The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ

[F15] For a nonnegative simple function $s = \sum_{ℓ = 1}^{m} a_{ℓ} 1_{E_{ℓ}}$ with disjoint measurable sets, its simple integral is $\sum_{ℓ} a_{ℓ} μ (E_{ℓ})$ , using $0 \cdot \infty = 0$ , and equals its nonnegative Lebesgue integral. In particular $\int 1_{X} d μ = μ (X)$ . The integral of a nonnegative simple function The nonnegative integral agrees with the simple integral on simple functions

[F16] Nonnegative integrals are monotone and satisfy $\int a f d μ = a \int f d μ$ for every finite $a > 0$ ; the zero function has integral zero. Monotonicity and nonnegative homogeneity of the nonnegative integral

[F17] A measure is finite when its total mass is finite, and sigma-finite when its space is a countable union of measurable sets of finite measure. In particular a finite measure is sigma-finite, using the whole space at every index. Finite, sigma-finite, and semifinite measures

[F18] For sigma-finite measure spaces $(X, A, μ)$ and $(Y, B, ν)$ , their product measure on $A \otimes B$ is defined by $(μ \times ν) (E) = \int_{X} ν (E_{x}) d μ = \int_{Y} μ (E^{y}) d ν$ , where $E_{x} = {y : (x, y) \in E}$ and $E^{y} = {x : (x, y) \in E}$ . The product measure of two sigma-finite measure spaces

Given: Countable Choice, the positive constant $c$ , the indices, Borel function, weighted-integrability hypothesis and zero extensions in the Statement.

Goal: Prove absolute integrability, the isotropic factor and the frequency-shell version.

1.1givenF12F13F14algebra

We first justify the measurability operations used below. If $f : X \to R^{m}$ and $g : X \to R^{n}$ are measurable on the same measurable space, the preimage of a rectangle under $(f, g)$ is the intersection of the two coordinate preimages. The target sets with measurable preimages form a sigma-algebra, so [F14] makes $(f, g)$ measurable into $R^{m + n}$ with its Borel sigma-algebra. By [F12] and [F13], composition with continuous multiplication, modulus, real and imaginary parts, or positive and negative parts therefore preserves measurability. The map $k \mapsto c ∣ k ∣$ is continuous from $R^{3} ∖ {0}$ to $(0, \infty)$ , and the formula for $Q_{i j}$ is continuous on that punctured space. Thus $v$ , $Q_{i j}$ and $v Q_{i j}$ are Borel there. For each Borel target set, the preimage under a zero extension is its punctured preimage, together with ${0}$ exactly when the target contains zero. By the trace assertion of [F13], and because the punctured space is open and ${0}$ is closed, this is Borel in $R^{3}$ . The functions $q_{i j}$ on $S^{2}$ are continuous, hence Borel. These observations also make all their moduli and real and imaginary parts, and the positive and negative parts of the latter, Borel. Euclidean Borel functions are measurable for the Lebesgue measures in use by [F13].

2.1givenF1F2F4F5F6F8F9F10F11F13F15F16F17step 1.1algebra

For $n \in S^{2}$ , $∣ q_{i j} (n) ∣ \leq 1$ and $\sum_{i = 1}^{3} q_{i i} (n) = 2$ . By [F1] and [F11] the sphere measure is finite. Facts [F15] and [F16] give $\int_{S^{2}} ∣ q_{i j} ∣ d σ \leq \int_{S^{2}} 1 d σ = σ (S^{2}) < \infty$ , so [F8] makes each $q_{i j}$ integrable. Summing [F4] over the three diagonal indices and using [F6] and [F15] gives $2 σ (S^{2}) = 8 π$ , hence $σ (S^{2}) = 4 π$ . The radial measure $d r$ restricted to $(0, \infty)$ is sigma-finite: the bounded measurable intervals $(0, m]$ , for integers $m \geq 1$ , have finite measure by [F9] and [F13] and cover the half-line. By [F17] this cover makes the radial measure sigma-finite, and the finite sphere measure is sigma-finite using the constant cover $S^{2}$ . Thus both factors required by [F2] and [F10] are sigma-finite.

3.1givenF1F3F5F7F8F15F16step 1.1step 2.1algebra

Let $h$ be a nonnegative Lebesgue measurable function on $(0, \infty)$ . By [F3] applied to the dilation and its inverse, $λ_{1} (c^{- 1} E) = c^{- 1} λ_{1} (E)$ for every Lebesgue measurable $E \subseteq (0, \infty)$ , and both images and preimages are measurable. For a nonnegative simple $s = \sum_{ℓ} a_{ℓ} 1_{E_{ℓ}}$ with disjoint measurable sets, [F15] therefore gives $\int_{0}^{\infty} s (c r) d r = \sum_{ℓ} a_{ℓ} λ_{1} (c^{- 1} E_{ℓ}) = c^{- 1} \int_{0}^{\infty} s (ω) d ω$ . Composition with the invertible dilation bijects the nonnegative simple minorants of $h$ with those of $h (c \cdot)$ ; taking suprema in [F7] proves $\int_{0}^{\infty} h (c r) d r = c^{- 1} \int_{0}^{\infty} h (ω) d ω$ . Take $h (ω) = ∣ u (ω) ∣ ω^{2}$ , which is measurable by step 1.1. Fact [F16] factors out $c^{- 2} > 0$ , giving $\int_{0}^{\infty} ∣ u (c r) ∣ r^{2} d r = c^{- 3} \int_{0}^{\infty} ∣ u (ω) ∣ ω^{2} d ω$ . Applying [F1] to $∣ v ∣$ , evaluating the inner constant integral with [F15] and [F16], and using step 2.1 now yields $\int_{R^{3}} ∣ v (k) ∣ d^{3} k = 4 π c^{- 3} \int_{0}^{\infty} ∣ u (ω) ∣ ω^{2} d ω < \infty$ . Since $∣ v Q_{i j} ∣ \leq ∣ v ∣$ , [F16] and [F8] establish integrability of both asserted functions.

4.1givenF1F2F3F4F6F8F10F12F14F15F16F18step 1.1step 2.1step 3.1algebra

On $(0, \infty) \times S^{2}$ , set $H (r, n) = u (c r) r^{2} q_{i j} (n)$ , using the product of the radial Lebesgue sigma-algebra and $B (S^{2})$ . Both factor measures are sigma-finite by step 2.1, so [F18] defines $d r \times σ$ on precisely this product sigma-algebra. The coordinate projections are measurable by [F14], since their preimages are rectangles. The factors of $H$ are compositions of these projections with the measurable radial and angular functions of step 1.1. Fact [F12] and the tuple and multiplication argument of step 1.1 therefore make $H$ product-measurable. By [F10], [F15], [F16] and the bounds and sigma-finiteness in steps 2.1 and 3.1, $\int ∣ H ∣ d (d r \times σ) \leq 4 π \int_{0}^{\infty} ∣ u (c r) ∣ r^{2} d r < \infty$ . Apply [F1] to the positive and negative parts of the real and imaginary parts of $v Q_{i j}$ , and recombine by [F8] and [F6]. Absolute convergence and [F2] justify the product and iterated integrations, giving $\int_{R^{3}} v Q_{i j} d^{3} k = \int_{0}^{\infty} u (c r) r^{2} (\int_{S^{2}} q_{i j} d σ) d r = (8 π δ_{i j} / 3) \int_{0}^{\infty} u (c r) r^{2} d r$ . For the complex substitution, apply step 3.1 to the positive and negative parts of $Re u$ and $Im u$ , each multiplied by $ω^{2}$ . Their integrals are finite by the weighted modulus hypothesis and [F16]. Since $r^{2} = c^{- 2} (c r)^{2}$ , [F16] and then [F8] and [F6] give $\int_{0}^{\infty} u (c r) r^{2} d r = c^{- 3} \int_{0}^{\infty} u (ω) ω^{2} d ω$ , proving the stated coefficient.

5.1givenF8F12F16step 1.1step 2.1step 3.1step 4.1∎

For a Borel $E \subseteq (0, \infty)$ , $u 1_{E}$ is Borel: for every Borel target set its preimage is $E \cap u^{- 1} (B)$ , together with $E^{c}$ exactly when $0 \in B$ . Moreover $∣ u 1_{E} ∣ ω^{2} \leq ∣ u ∣ ω^{2}$ , so its weighted absolute integral is finite by [F16]. Steps 1.1, 2.1, 3.1 and 4.1 apply with this function and its zero extension. This proves the claimed frequency-set identity, in particular the frequency-shell identity.

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Isotropic thermal bath time correlator

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