Invertible Complex Measures on Euclidean Spaces

Alexander Lindner; David Berger

Invertible Complex Measures on Euclidean Spaces

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Pith's one-line read A complex measure on R^n is convolution-invertible exactly when it factors as a shift, finitely many copies of a fixed one-dimensional signed measure on lines through the origin, and the exponential of a complex measure.

desk verdict A complete, carefully proved characterization of invertible complex measures on R^n; the central argument is conditional on Taylor's spectral-stability results, but the dependency is explicit and legitimate. read the letter →

arxiv 2506.09012 v2 pith:ZITQFQPT submitted 2025-06-10 math.PR math.FA

David Berger , Alexander Lindner This is my paper

classification math.PRmath.FA MSC 43A0543A2060E1060E07

keywords invertiblecomplexmeasuresconvolutioncharacteristicfunctionsgeneralizedFourierseriesquasi-infinitelydivisibledistributionsLévy–KhintchinerepresentationBanachalgebravaluedmeasure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a complete structural characterization of invertible complex measures on Euclidean space. It shows that a complex measure on R^n is invertible under convolution exactly when it can be built from a shift, finitely many copies of a single fixed one-dimensional signed measure placed on lines through the origin, and the exponential of a complex measure. The same description yields a Lévy–Khintchine type formula for the characteristic function of every such measure, with the non-finite part always supported on finitely many lines with a fixed radial density. This settles the multidimensional analogue of a problem solved in one dimension in 1971. In fact, the multidimensional structure is no more complicated than the one-dimensional one.

What carries the argument

The proof is carried by three components. First, a structural decomposition theorem for invertible measures on locally compact abelian groups reduces any invertible µ to a convolution of factors each consisting of a point mass plus a density with respect to Haar measure on R^q × $R_d^{{n-q}}$. Second, the fixed signed measure σ, with characteristic function (1+iz)/(1-iz), is the single building block that accounts for the non-exponential part; its characteristic function satisfies exp(∫($e^{{ixz}}$-1) m $e^{{-|x|}}$/x dx) = ($σ^{{∧}}$(z))^m, which is used to rewrite the σ-factors as Lévy measures on lines. Third, for the intermediate dimensions q ∈ {1,...,n-1}, the paper develops a theory of Banach algebra-valued generalized Fourier series and a distinguished logarithm for functions taking values in a semisimple commutative unital Banach algebra with connected Gelfand space, which lets it characterize when the characteristic function of such a factor is invertible in the algebra GFS($R^{{n-q}}$, A).

What would settle it

The claim would be falsified by an invertible complex measure on $R^{2}$ whose characteristic function, under the Lévy–Khintchine representation of Theorem 2.4, requires the singular measure Λ on the unit sphere to be supported on infinitely many points; the theorem asserts Λ is always finitely supported.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 2.1: a complex measure µ on R^n is convolution-invertible if and only if it can be written as δ_γ^n * U_1($σ^{{*m_1}}$ ⊗ $δ_0^{{n-1}}$) * ... * U_p($σ^{{*m_p}}$ ⊗ $δ_0^{{n-1}}$) * exp(ν), where σ is the fixed signed measure whose characteristic function is (1+iz)/(1-iz), the U_j are orthogonal matrices, m_j are integers, γ is a shift vector, and ν is a complex measure. Equivalently, at the level of characteristic functions, µ is invertible exactly when its Fourier transform has the form c exp(iγ·z + ∫($e^{{iz·x}}$-1)(ν_0+ν_1)(dx)) with ν_0 finite and ν_1 given by $r^{{-1}}$$e^{{-r}}$ dr times a finite signed measure on the unit sphere, so the non-finite part lives only on finitely many lines through the origin. When n=1 the theorem reduces to the 1971 one-dimensional characterization.

Load-bearing premise

The argument rests on the assumption that the inverse of each factor that is a point mass plus a density on R^q × $R_d^{{n-q}}$ stays in the same subalgebra and that invertibility in that subalgebra is equivalent to invertibility in the full measure algebra M(R^n); if that spectral-stability input failed for some q, the chain of reasoning would break.

Editorial extensions

If this is right

Every invertible complex measure on R^n has a Lévy–Khintchine representation with a complex Lévy type measure that integrates min(1,|x|).
For absolutely continuous-plus-point-mass measures and for discrete measures, invertibility is equivalent to invertibility of all one-dimensional projections.
An invertible finite signed measure can be factored with a signed measure ν and the normalization factor µ(R^n), and its inverse is again a signed measure.
An invertible complex measure has convolution roots of all orders if and only if it is a shift times an exponential, i.e. the σ-factors are absent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the non-finite part of the Lévy measure is always supported on finitely many lines, invertibility in M(R^n) is far more rigid than the mere non-vanishing of the characteristic function; this suggests that the failure of the naive condition inf |Xhat µ| > 0 to be sufficient is already captured by line-like obstructions.
The Banach algebra valued generalized Fourier series machinery could transfer to other settings where one studies invertibility of functions with values in a commutative Banach algebra, such as convolution algebras on products of Euclidean and discrete abelian groups.
The paper's description of invertible measures gives an explicit way to construct probability measures that are quasi-infinitely divisible but not infinitely divisible: take a probability measure equal to δ_γ * σ^{*u} * exp(ν) with u ≠ 0 and normalize; the signed density e^{-r}/r along a line yields the quasi-Lévy measure.
One could test whether the finiteness of the support of Λ on the sphere is genuinely necessary by trying to construct an invertible measure whose characteristic function requires a continuum of lines; Theorem 2.4 rules that out, so a positive example would refute the theorem as stated.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this is a major result. Berger and Lindner give the full analogue of Taylor's 1971 theorem for R^n, and the answer is strikingly clean: every invertible complex measure is a shift, a convolution of finitely many line measures U_i(σ^{*m_i} ⊗ δ_0^{n-1}), and exp(ν). They also get the Lévy–Khintchine type representation and several Cramér–Wold devices. The n-dimensional theorem is new, and the intermediate cases in Taylor's factorization are genuinely hard. The Banach algebra machinery they build—GFS with values in a semisimple commutative unital algebra with connected Gelfand space, the distinguished logarithm theorem, and the invertibility characterization in Theorem 7.9—is a real contribution. The paper is careful about the topology classification and Haar measures along the way.

The main theorem is downstream of Taylor's Theorem 1.2 and Proposition 4.3. Section 8 leans on Proposition 4.3 to know that the inverse of a measure in Cδ_0 + L(R^q) ⊗ discrete stays in the same subalgebra, which lets the inverse's characteristic function land in GFS(R^{n-q}, A). If that statement failed for some q, the proof of Theorem 8.1 would collapse. But it is a published theorem of Taylor, and the paper flags it as nontrivial. I do not count this as a flaw; it is a normal external dependency. The stress-test note correctly identifies the load-bearing point, but it is not an internal gap. A referee should verify that Proposition 4.3 covers every q as stated, since the paper's interpretation of spectral stability is exactly what makes the GFS inversion step go through.

This paper is for specialists in measure algebras, convolution operators, and quasi-infinitely divisible distributions. It deserves a serious referee. The approach is sound, the new GFS results look correct, and the main theorem is important. I would send it to peer review without hesitation. If I were refereeing, I would ask for a short remark making the dependence on Proposition 4.3 impossible to miss.

Referee Report

0 major / 4 minor

Summary. The paper characterizes convolution-invertible complex measures on R^n. The main theorem (Theorem 2.1) states that µ ∈ M(R^n) is invertible if and only if it admits a factorization µ = δ_γ^n * U_1(σ^{*m_1} ⊗ δ_0^{n-1}) * ... * U_p(σ^{*m_p} ⊗ δ_0^{n-1}) * exp(ν), where σ is the one-dimensional signed measure with characteristic function (1+iz)/(1-iz), U_i are orthogonal matrices, m_i are integers, γ ∈ R^n, ν ∈ M(R^n), and p ∈ N_0. This directly extends Taylor's 1971 one-dimensional characterization. The authors also give an equivalent characteristic-function formulation (Theorem 2.4) with a complex Lévy-type measure, and a version for finite signed measures (Corollary 2.2). The proof uses Taylor's general decomposition theorem for invertible measures on locally compact abelian groups (Theorem 1.2), Hewitt's classification of group topologies on R^n (Lemma 4.1), and a substantial new development of Banach-algebra-valued generalized Fourier series, including a distinguished logarithm theorem for semisimple commutative unital Banach algebras with connected Gelfand space and an invertibility criterion for GFS(R^n, A) (Theorems 7.4, 7.7, 7.9). These tools are applied in Section 8 to the intermediate-dimensional case 1 ≤ q ≤ n-1, which is the core technical novelty.

Significance. If the main theorem is correct, it provides a complete and surprisingly simple structural description of all convolution-invertible complex measures on R^n, showing that only one-dimensional σ-factors, shifts, and exponentials are needed. The paper is well structured and carefully distinguishes its own contributions from deep external results (Taylor's measure-algebra spectral theory, Hewitt's classification), with Proposition 4.3 explicitly cited as nontrivial and load-bearing. The new Banach-algebra machinery for generalized Fourier series is interesting in its own right and likely to be useful beyond this paper. The applications to quasi-infinitely divisible distributions and to convolution roots of all orders (Theorem 10.1) are natural and credibly derived. The proofs are detailed and internally consistent; the main theorem is conditional on Taylor's spectral-stability results, but those are standard published theorems and the dependence is clearly flagged rather than hidden.

minor comments (4)

[Theorem 5.3] In condition (v'), the notation λ^1 should be λ^n: the measure ν is on R^n, so the density g should be in L^1(R^n) and the expression should read ν(dx) = g(x) λ^n(dx). The same typo appears in the proof sentence 'µ = exp(gλ1 + βδ_0^n)'.
[Proposition 4.2] The indexing in the statement is incorrect: the functions f_i are listed for i = 1, ..., q, but there are p factors, so it should read i = 1, ..., p.
[Theorem 2.1] The statement contains a duplicated 'Let n ∈ N': it appears once before the definition of σ and again immediately after.
[Theorem 10.1] In the proof of (i) implies (iv), the assertion that the non-integer coefficients m_j/k cannot be absorbed into a different representation is described as 'easily seen'; a short justification via mutual singularity of the measures τ_{u_j} (supported on distinct lines) and the finiteness of ν_0 would make the argument more transparent.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the n-dimensional invertibility characterization is derived from Taylor's LCA theorem and independent case analyses; the only self-citation (Berger [4]) is a published tool rather than a load-bearing reduction.

full rationale

The paper's central claim (Theorem 2.1) is not assumed as an input. Its 'only if' direction starts from Taylor's Theorem 1.2 (a published LCA-group decomposition) and determines the invertible factors case by case: q = n in Section 5, q = 0 in Section 6, and q in {1,...,n-1} in Sections 7-8. The intermediate case is handled by Banach-algebra-valued generalized Fourier series (Theorems 7.7 and 7.9), whose proofs are internal except for the scalar discrete-measure result of Alexeev-Khartov [1, Thm. 3.2], which is external. The load-bearing spectral-stability statement, Proposition 4.3, is explicitly attributed to Taylor [28, Thm. 3.3 with Prop. 4.1] and is not proved in the paper; this is a genuine external dependency and a conditional-correctness risk, but it is not circular. The only self-citation of note is Berger [4, Thm. 4.4] in Proposition 5.2, used to obtain the CLK1_0 form (v) for measures alpha*delta0 + f*lambda on R. That theorem is a published, peer-reviewed result from 2019 and is used as a tool; moreover, the sigma^m representation needed for the main theorem in the q=1 case is ultimately Taylor's Theorem 1.1, which is stated as an external anchor. No fitted parameter is relabeled as a prediction, no known result is merely renamed, and no uniqueness theorem of the present authors is invoked to force the choice of sigma. Hence the derivation is self-contained against external benchmarks apart from the named Taylor/Alexeev-Khartov inputs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted constants or invented entities appear; the paper is a theorem paper. The listed axioms are the unproved external results on which the proof depends, chiefly Taylor's factorization theorem, Hewitt's classification of group topologies, the spectral stability of the subalgebra Cδ_0 + L(R^q) ⊗ discrete, and the identification of Gelfand spaces of projective tensor products.

assumptions (6)

domain assumption Taylor's Theorem 1.2 (cited): every invertible µ factorizes as µ_1*...*µ_p*exp(ν), where each µ_i = α_i δ_0 + regular measure absolutely continuous w.r.t. Haar measure on some R^{q_i} × R_d^{n-q_i}.
Stated as Theorem 1.2 without proof; the paper's entire reduction strategy depends on this factorization.
domain assumption Hewitt's classification (cited): any locally compact abelian group topology on R^n finer than Euclidean is topologically isomorphic to R^q × R_d^{n-q}.
Used in Lemma 4.1 to enumerate all possible topologies τ_i arising in Taylor's theorem.
domain assumption Taylor [28] spectral stability (cited in Prop. 4.3): for µ = U(α δ_0 + f λ^q ⊗ ζ_C^{n-q}), the inverse has the same form and the spectrum in the subalgebra equals the spectrum in M(R^n).
Critical in Section 8: guarantees that the inverse's characteristic function is again a generalized Fourier series with coefficients in the Banach algebra A.
domain assumption Alexeev-Khartov [1] and Taylor [27] (cited, q=0 case): discrete µ is invertible iff inf_z |bµ(z)| > 0 iff µ = δ_γ * exp(ν') with discrete finite ν'.
Used in Theorem 6.1 to handle the purely discrete factors in the decomposition.
standard math Kaniuth [12] (cited): for unital commutative Banach algebras, Δ(L^1(R_d^n) ⊗π A) ≅ Δ(L^1(R_d^n)) × Δ(A).
Textbook tensor product Gelfand space identification used in Theorem 7.7 to characterize invertibility in GFS(R^n, A).
domain assumption Berger [4, Thm. 4.4] (cited, n=1 case): invertible α δ_0 + f λ admits CLK_1^0 with Lévy measure m e^{-|x|}/x dx + g dx.
Self-cited published theorem used in Proposition 5.2 for the one-dimensional absolutely-continuous-plus-point-mass case.

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Pith. "Pith review of Invertible Complex Measures on Euclidean Spaces." pith.science (2026). https://pith.science/paper/ZITQFQPT

@misc{pith2026250609012,
  author       = {Pith},
  title        = {Pith review of: Invertible Complex Measures on Euclidean Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZITQFQPT}},
  note         = {Machine review of arXiv:2506.09012}
}

abstract

In 1971 Taylor characterised all complex measures on $\mathbb{R}$ that are invertible with respect to convolution as those which can be written in the form $\delta_\gamma \ast \sigma^{\ast m} \ast \exp(\nu)$ for some $\gamma\in \mathbb{R}$, some complex measure $\nu$, some $m\in \mathbb{Z}$ and a given fixed invertible finite signed measure $\sigma$ (which has characteristic function $\mathbb{R} \ni z \mapsto (1+i z)/(1-i z)$). We extend Taylor's result to complex measures on $\mathbb{R}^n$. Somewhat surprisingly, the structure of invertible complex measures on $\mathbb{R}^n$ is not much more complicated than that of complex measures on $\mathbb{R}$, in the sense that they can be represented as $\delta_\gamma \ast \sigma_1^{\ast m_1} \ast \ldots \ast \sigma_p^{\ast m_p} \ast \exp(\nu)$ for some $\gamma \in \mathbb{R}^n$, some complex measure $\nu$ and $m_1,\ldots, m_p\in \mathbb{Z}$, where the $\sigma_i$ correspond to $\sigma$ in the one-dimensional case and actually live on $1$-dimensional subspaces of $\mathbb{R}^n$. Our proof relies on a general result of Taylor for invertible complex measures on locally compact abelian groups. To apply Taylor's result, we extend some existing results for $\mathbb{C}$-valued functions to functions with values in a semisimple commutative unital Banach algebra with connected Gelfand space. The study of invertible complex measures on $\mathbb{R}^n$ has some impact on the theory of quasi-infinitely divisible probability distributions on $\mathbb{R}^n$.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Alexeev and A

I. Alexeev and A. Khartov, A criterion and a Cram´ er-Wold device for quasi-infinite divisibility for discrete multivariate proability laws. Electron. J. Probab. 28, paper number 130, 1-17, 2023

work page 2023

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work page 2023

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work page 2011

[4] [4]

Berger, On quasi-infinitely divisible distributions with a point mass

D. Berger, On quasi-infinitely divisible distributions with a point mass. Math. Nachr. 292, 1674–1684, 2019

work page 2019

[5] [5]

Berger and M

D. Berger and M. Kutlu, Quasi-infinite divisibility of a class of distributions with discrete part. Proc. Amer. Math. Soc. 151, 2211–2224, 2023

work page 2023

[6] [6]

Berger, M

D. Berger, M. Kutlu and A. Lindner, On multivariate quasi-infinitely divisible distributions. In: A lifetime of excursions through random walks and L´ evy processes. A volume in honour of Ron Doney’s 80th birthday . Progress in Probability 78, Birkh¨ auser, Cham, 2021

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