The structure of 𝓐-free measures revisited

D. Mitrovic; Dj. Vujadinović

doi:10.1515/anona-2020-0223

Article Open Access

The structure of 𝓐-free measures revisited

D. Mitrovic and Dj. Vujadinović

Published/Copyright: June 10, 2020

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From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

Abstract

We refine a recent result on the structure of measures satisfying a linear partial differential equation 𝓐μ = σ, μ, σ are Radon measures, considering the measure μ(x) = g(x)dx + μ_us(x̃)(μ_s(x̄) + dx̄) where x = (x̃,x̄) ∈ ℝ^k × ℝ^d−k, μ_us is a uniformly singular measure in x̃₀ and the measure μ_s is a singular measure. We proved that for μ_us-a.e. x̃₀ the range of the Radon-Nykodim derivative f~(x~0)=dμusd|μus|(x~0) is in the set ∩_ξ̃∈P̃𝓚erA^P̃(ξ) and, if μ_s is different to zero, for μ_s-a.e. x̄₀ the range of the Radon-Nykodim derivative f¯(x¯0)=dμsd|μs|(x¯0) is in the set ∪_ξ̄∈P̄ 𝓚erA^P̄(ξ) where P̃ × P̄ = P is a manifold determined by the main symbol A^P = A^P̃ ⋅ A^P̄ of the operator 𝓐.

Keywords: Structure of measures

MSC 2010: 35D30

1 Introduction

In the paper, we consider a finite Radon measure μ = (μ₁, …, μ_m) defined on ℝ^d satisfying the system of partial differential equations

Aμ=∑α∈I∂α(Aαμ)=0inD′(Rd;Rn) (1.1)

where I = I₁ × I₂ × … × I_n ⊂ {α = (α₁, …, α_d) : α_s ∈ ℕ∪ {0}, s = 1, …, d}ⁿ is a set of multi-indexes, ∂α=∂x1α1∂x2α2…∂xdαd, and A_α : ℝ^d → M^n×m are smooth mappings from ℝ^d into the space of real n × m matrices. Written coordinate-wise, we actually have the following system of equations

Ajμ=∑α∈Ij∑r=1m∂α(ajrαμr)=0,j=1,…,n, (1.2)

where I_j ⊂ {α = (α₁, …, α_d) : α_s ∈ ℕ∪ {0}, s = 1, …, d}. Denote by A_j, j = 1, …, n, the principal symbol of the operator 𝓐_j given by

Aj(x,ξ)=∑α∈Ij′∑r=1majrα(x)(2πiξ)α,Ij′⊂Ij. (1.3)

The sum given above is taken over all terms from (1.2) whose order of derivative α is not dominated by any other multi-index from I_j. As usual, ξα=ξ1α1…ξdαd for α = (α₁, …, α_d), and ∣α∣ = α₁ + … + α_d.

For instance, for the (scalar) operator 𝓐 = ∂_x₁ + ∂_x₂ + ∂x22, we have I = I₁ = {(1, 0), (0, 1), (0, 2)} and I′ = I′₁ = {(1, 0), (0, 2)}.

Let us emphasize the fact that the equation (1.1) includes the case

Aμ=σ, (1.4)

where σ ∈ 𝓜(ℝ^d, ℝⁿ). Namely, regarding the equation (1.1) we may consider the measure μ̃ = (μ, σ) ∈ 𝓜(ℝ^d, ℝ^m+n) and the equation 𝓐̃μ̃ = 0 (where 0’th-order term was added to 𝓐̃) which is equivalent to (1.4).

We are interested in the range of the Radon-Nikodym derivatives f~(x~)=dμusd|μus|(x~) and f¯(x¯)=dμsd|μs|(x¯) where the measures μ_us and μ_s are the parts of the measure

μ=g(x)dx+μus(x~)(μs(x¯)+dx¯),g∈L1(Rd), (1.5)

satisfying (1.1). The measure μ_s is a singular measure while μ_us is uniformly singular measure. Roughly speaking, we require that μ_s is singular with respect to every of the variables. For instance, such a condition is not fulfilled by the measure μ = δ(x₁ − x₂)dx₂ since it is not singular with respect to x₂. However, if we introduce the change of variables z₁ = x₁ − x₂ and z₂ = x₂, then μ becomes δ(z₁)dz₂ and we have the combination of the uniformly singular measure and a regular measure.

Definition 1

We say that the measure μ_s is uniformly singular if for μ_s-almost every x ∈ ℝ^d there exist real positive functions α(ε), β(ε), ε ∈ ℝ, satisfying α(ε)ε→0,εβ(ε)→0, and a family of balls B(x, α(ε)) such that limε→0μs(B(x,β(ε))∖B(x,α(ε)))|μs|(B(x,α(ε)))=0.

For instance, it is clear that the measure δ(x₁)δ(x₂) satisfies the latter condition with E_ε = {(0, 0)} and arbitrary α(ε) and β(ε) satisfying α(ε)ε→0 and εβ(ε)→0. In general, a measure supported on the set whose Hausdorf dimension is less than n − 1 is a candidate for the uniformly singular measure (see below for further explanations). It is not difficult to see that δ(x₁)dx₂ is a singular measure on ℝ², but not uniformly singular.

If we put k = 0 i.e. m = d (implying that we do not have uniformly singular part: μ = g(x) dx + μ_s) then the problem is solved elegantly in [3] confirming the conjecture from [1] that for the k-th order operator 𝓐, the function f¯(x)=dμsd|μs|(x) must take values in the wave cone Λ_𝓐 = ∪_∣ξ∣=1 𝓚erA^k(ξ) where A^k(ξ) is the sum of all symbols of order k (see [3] for details and thorough information concerning history and applications of this issue; in particular in the calculus of variations and geometric measure theory).

In the case when μ_us is nontrivial, we are able to prove a stronger result as announced in the abstract and formally introduced in Theorem 2. However, in the latter case, we have the measure of very special form which actually separates variables. This shortens space of measures that fit into our considerations, but the space is far from trivial. For instance, consider the singular measure μ = δ(x₁ − x₂) dx₂. This measure is not uniformly singular and it satisfies the equation

∂x1μ+∂x2μ=0

and after introducing the change of variables x₁ − x₂ = y₁ and x₂ = y₂ we reduce the measure μ on the form (1.5). Also, the (n − 1)-dimensional Hausdorff measure that can be locally represented in the from δ(x₁ − g(x₂, …, x_d)) dx₂ … dx_d and, after the change of variables z₁ = x₁ − g(x₂, …, x_d), x_j = z_j, j = 2, …, d, it gets the form (1.5). Moreover, if we augment (1.1) with initial conditions involving an uniformly singular measure, then, at least in the case of first order scalar equations, the solution will contain the uniformly singular measure as well (since the solution is given along characteristics).

To continue, we assume that all the principal symbols A_j, j = 1, …, n, can be represented in the form

Aj(x,ξ)=A~j(x~,ξ~)⋅A¯j(x¯,ξ¯). (1.6)

For instance, the matrix A_j can be of the form ξ₁ ξ₂ where Ã_j(ξ̃) = ξ₁ and Ā_j(ξ̄) = ξ₂. The operator determined by such a matrix A₁ is actually a hyperbolic operator.

The latter restriction essentially means that we can separate variables x̄ and x̃ while for the dual variables ξ, this means that we first move in the direction of ξ̄ determined by the matrix Ā_j, and then in the direction ξ̃ determined by the matrix Ã_j.

Next, we assume that there exist multi-indices βj=(β1j,⋯,βkj)⊂N+k (depending on j = 1, …, n) and β = (β_k+1, ⋯, β_d) ⊂ N+d−k (independent of j = 1, …, n), (β1j,⋯,βkj,βk+1,…,βd)∈Ij′ such that for any positive λ ∈ ℝ the following homogeneity assumption holds for every j = 1, …, n:

A~j(x~,λβ1jξ1,…,λβkjξk)=λA~j(x~,ξ~),A¯j(x¯,λβk+1ξk+1,…,λβdξd)=λA¯j(x¯,ξ¯) (1.7)

implying that

αrjβrj=1for every r∈1,…,k andαrβr=1for every r∈k+1,…,d. (1.8)

We then introduce the homogeneity manifolds:

P~j={ξ~∈Rk:|ξ1|1/β1j+⋯+|ξk|1/βkj=1}P¯={ξ¯∈Rd−k:|ξk+1|1/βk+1+⋯+|ξd|1/βd=1}

and the corresponding projections

πj(ξ~)=ξ1|ξ1|1/β1j+⋯+|ξk|1/βkjβ1j,…,ξk|ξ1|1/β1j+⋯+|ξk|1/βkjβkj,ξ~∈Rkπ(ξ¯)=ξk+1|ξk+1|1/βk+1+⋯+|ξd|1/βdβk+1,…,ξd|ξk+1|1/βk+1+⋯+|ξd|1/βdβd,ξ¯∈Rd−k.

Finally, we can formulate the main theorem of the paper.

Theorem 2

Let μ be a solution to (1.1) of the form (1.5). Then, for ∣μ_us∣-almost every x̃ ∈ ℝ^k and ∣μ^s∣-almost every x̄ ∈ ℝ^d−k there exists ξ̄ ∈ P̄ such that:

∑α∈Ij′∑l=1majlα(x)(2πiξ)αf~l(x~)f¯l(x¯)=0,Hk−1−a.e.ξ~∈P~j,j=1,…,n. (1.9)

If for all j = 1, …, n, the manifolds P̃_j would be the same, say P̃, and we have the same set of dominating multi-indices I′ = I′_j, j = 1, …, n, then we could rewrite (1.9) in the form

f~(x~)f¯(x¯)∈∩ξ~∈P~∪ξ¯∈P¯Ker∑α∈I′(2πiξ)αAα(x)forμus⊗μsa.e.x∈Rd (1.10)

for appropriate matrices A_α, α ∈ I′. If we do not have the uniformly singular part and P̄ is the sphere in ℝ^d, then (1.10) is actually the statement of the main result from [3]. In their elegant proof, the main tool was the concept of tangent measures in the sense of [7]. We will pursue this approach here as well but with slightly more refined arguments which take into account properties of the measure (splitting on singular and uniformly singular part) as well as properties of the operator 𝓐 itself (principal symbols do not have to be defined on a sphere).

We will dedicate the last section to the proof of the theorem. In the next section, we shall prove it in the case of first order constant coefficient operators and the scalar measure which captures all the elements of the general situation. The proof is based on the blow up method [8] (which naturally leads us to the tangent measures) and appropriate usage of Fourier multiplier operators (as in deriving appropriate defect functionals [2, 5]).

Let us recall that the Fourier multiplier operator T_ψ with the symbol ψ is defined via the Fourier and inverse Fourier transform

Tψu(x)=F−1(ψ(ξ)F(u))(x),u∈L2(Rd),

where the Fourier and the inverse Fourier transforms are given by

F(u)(ξ)=u^(ξ)=∫e−2πix⋅ξu(x)dx,F−1(u)(x)=uˇ(x)=∫e2πix⋅ξu(ξ)dξ.

For properties of the Fourier multiplier operators one can consult [4].

As for the tangent measure, we shall use the property of any locally finite measure [7, Lemma 2.4] and to recall the following theorem (see also [7, Theorem 2.5]) representing a special case of the tangent measures.

Theorem 3

For any locally finite measure μ defined on ℝ^d and any multi-index (α₁, …, α_d) ∈ ℕ^d and μ-almost every x = (x₁, …, x_d) ∈ ℝ^d there exists a locally finite measure ν such that along a subsequence as r → 0 it holds for every φ ∈ C₀(ℝ^d)

∫φ(y)dμ(x1+rα1y1,…,xd+rαdyd)μ((x1−rα1,x1+rα1)×⋯×(xd−rαd,xd+rαd))→∫φ(y)dν(y)asr→0.

The paper is organized as follows. In the next section, we shall prove the theorem in the case μ = μ_us(x̃) dx̄ which contains arguments concerning uniformly singular measures which are new with respect to the ones from [3]. In the last section, we prove the result in the full generality.

2 Proof of Theorem 2 in the case of the hyperbolic constant coefficients operator

Here, we shall prove Theorem 2 when the scalar finite Radon measure μ ∈ 𝓜(ℝ^d) of the form μ(x) = μ_us(x̃)dx̄ satisfies the equation

∑|α|=m∂x~α(aαμ)+a0σ=0, (2.11)

where a_α are constants and σ is also finite scalar Radon measure. The proof is essentially the same for the general operator of the form given in (1.1), but it is a bit less technical for (2.11). The proof in full generality is given in the next section.

Before we start, let f̃(x₀) be the Radon-Nykodim derivative of μ_us with respect to ∣μ_us∣:

f~(x~0)=dμusd|μus|(x~0).

We fix a convolution kernel ρ : ℝ → ℝ which is a smooth, compactly supported function of total mass one and convolve (2.11) by

1εkρ(x~ε)=1εkΠj=1kρ(xjε)=1εkΠj=1kρj,ε(xj)=ρϵ(x).

Then, we take an arbitrary φ ∈ Cc1 (ℝ^k) and φ₁ ∈ Cc1 (ℝ^d−k) and test the convolved equation on the product of such functions. We get (below, we denote μusε = μ_us⋆ ρ_ε and 〈φ(ỹ), μ_us(ỹ)〉 = ∫_ℝ^k φ(ỹ)dμ_us(ỹ))

∑|α|=maα∫Rd〈1εkΠj=1kρ(xj−yjε),μus(y~)〉∂x~αφ(x~)φ1(x¯)dx+a0∫Rdφ(x~)φ1(x¯)dσε(x)=0. (2.12)

We now fix x̃₀ ∈ ℝ^k and take φ(x~−x~0ε) instead of φ in (2.12). We get (below, w = (w̃,w̄)):

ϵ−m∑|α|=maα∫Rd〈1εkΠj=1kρ(xj−yjε),μus(y~)〉∂w~αφ(w~)|w~=x~−x~0εφ1(w¯)dw+a0∫Rdφ(x~−x~0ε)φ1(x¯)dσε(x)=0. (2.13)

We now introduce in the first integral above the change of variables x̃ = x̃₀ + ε w̃ and multiply the entire expression by ε^m. We get (we denote x̃₀ = (x10,…,xk0)) and w̃ = (w₁, …, w_k)):

∑|α|=maα∫Rd〈Πj=1kρ(xj0−yjε+wj),μus(y~)〉∂w~αφ(w~)φ1(w¯)dw+εma0∫Rdφ(x~−x~0ε)φ1(x¯)dσε(x)=0. (2.14)

We consider separately terms involving the measure μ_us:

Mus,ε:=∫Rk〈Πj=1kρ(xj0−yjε+wj),μus(y~)〉aα∂wαφ(w~)dw~, (2.15)

We rewrite (2.15) in the form (for α and β given in the uniform singularity definition)

Mus,ε=∫Rk(∫B(x~0,α(ε))+∫B(x~0,β(ε))∖B(x~0,α(ε))+∫Rk∖B(x~0,β(ε))Πj=1kρ(xj0−yjε+ωj)dμus(y~))×aα∂wαφ(w~)dw~. (2.16)

Now, according to the assumptions for the uniformly singular measures (see Definition 1) and the fact that ρ is compactly supported:

μus(B(x~0,β(ε))∖B(x~0,α(ε)))|μus|(B(x~0,α(ε))→0asε→0;ρ(x~0−yε+w~)→0,y∉B(x~0,β(ε))and, obviouslyρ(x~0−yε+w~)→ρ(w~),y∈B(x~0,α(ε))

we get after dividing (2.16) by ∣μ_us∣(B(x̃₀, α(ε)) and letting ε → 0 in (2.16) (for ∣μ_us∣-a.e. x̃₀ ∈ ℝ^d):

limε→0Mus,ε|μus|(B(x~0,α(ε))=f~(x~0)∫Rkρ(w~)aα∂wαφ(w~)dw~. (2.17)

If we take here

φ(w~)=Tψ(ξ~)|ξ~|m(ρ(w~))¯, (2.18)

where Tψ(ξ~)|ξ~|m is the Fourier multiplier operator with the symbol ψ(ξ~)|ξ~|m, we find after taking the Plancherel theorem into account:

limε→0Mus,ε|μus|(B(x~0,α(ε))=f~(x~0)∫Rk|ρ^|2(ξ~)aα(iξ~)α|ξ~|mψ(ξ~)dξ~. (2.19)

From here, we conclude after letting ε → 0 in (2.14) with φ given by (2.18)

f~(x~0)∑|α|=m∫Rd|ρ^|2(ξ~)aα(iξ~)α|ξ~|mψ(ξ~)φ1(x¯)dξ~dx¯=0. (2.20)

From here, due to arbitrariness of ψ, we conclude that (1.10) holds (without the ∪ sign since we do not have the singular part, but only the uniformly singular part of the measure).

3 Proof of Theorem 2; general case

In this section, we consider equations (1.2) under the assumption μ(x) = μ_us(x̃) μ_s(x̄). We have omitted the terms g(x) dx and μ_us(x̃) dx̃ appearing in (1.5) since for the purely Lebesgue part there is nothing to prove and the term μ_us(x̃) dx̃ is handled as the μ(x) = μ_us(x̃) μ_s(x̄) by replacing μ_s(x̄) by dx̄.

We shall rewrite in the form:

∑α∈I~j,β∈I¯∑r=1m∂x~α∂x¯β(ajrαβ(x)μusr(x~)μsr(x¯))+Alowerjμ=0 (3.21)

where Ĩ_j and Ī are set of indexes corresponding to the principal symbol of the operator 𝓐.

We shall prove Theorem 2 by following the steps from the previous section together with the approach taken in [3] and we refer the reader there for clarifications.

We start by fixing j in (1.2) and the convolution kernel ρ : ℝ → ℝ which is smooth, compactly supported with total mass one. We then denote

ρj,ε(x~)=1εβ1j+⋯+βkj∏s=1kρ(xsεβsj)andρ(w~)=∏s=1kρ(ws),

and convolve (3.21) by ρ_j,ε. We have for (ajrαβμus)ε=(aαβjrμus)⋆ρj,ε(x~):

∑α∈I~j,β∈I¯∑r=1m∂αβ((ajrαβμusr)εμsr+Alowerjμ⋆Πr=1kρj,ε(x~)=0. (3.22)

We then apply a test function φ ∈ Cc∞ (ℝ^k) on (3.22) to get

∑α∈I~j,β∈I¯(−1)|α|∑r=1m∂β∫Rk〈ρj,ε(x~−y~),ajrαβ(y~,x¯)μusr(y~)〉∂αφ(x~)dx~dμsr(x¯)+Alowerjμ⋆ρj,ε(x~),φ(x~)=0. (3.23)

Now, we fix z = (z̃,z̄) ∈ ℝ^d and take

φε(x~−z~)=φ(x1−z1εβ1j,…,xk−zkεβkj)

in (3.23) instead of φ. At the same time, for the variable x̄, we fix z = (z̃,z̄) ∈ ℝ^d we introduce the change x̄ = (z_k+1 + ε^β_k+1w_k+1, …, z_d + ε^β_d w_d) = z̄ + ε^β̄ w̄ in (3.23) and apply the test function φ₁(w̄). Multiplying the obtained expression by ε and taking into account (1.8), we conclude

∑α∈I~j,β∈I¯(−1)|α|+|β|∑r=1m∫Rd〈ρj,ε(x~−y~),ajrαβ(y~,x¯)μusr(y~)〉∂αφε(x~−z~)∂βφ1(w¯)dμsr(z¯+εβ¯w¯)dx~++ε〈Alowerjμ⋆ρj,ε(x~),φφ1〉=0. (3.24)

Next, we introduce the change of variables x~=(z1+εβ1jw1,…,zk+εβkjwk) in the first term on the left-hand side of (3.24). We get

∑α∈I~j,β∈I¯(−1)|α|+|β|∑r=1m∫Rd〈∏s=1kρ(zs−ysεβsj+ws),(ajrαβ(y~,x¯)μusr(y~)〉∂αφ(w~)∂βφ1(w¯)dμsr(z¯+εβ¯w¯)dw+Rε=0, (3.25)

where

Rε=ε〈Alowerjμ⋆Πr=1kρj,ε(x~),φφ1〉→0

by definition of the principal symbol. Now, we divide (3.25) by

|μus|(B(0,α(ε)))|μs|((zk+1−rα1,xk+1+rα1)×⋯×(zd−rαd,xd+rαd))

where α(ε) is given in the definition of the uniform singularity and let ε → 0. Taking into account the uniform singularity assumptions as in (2.16) and Theorem 3, we get

∑α∈I~j,β∈I¯(−1)|α|+|β|∑r=1majrαβ(z)∫Rdρ(w~)∂w~αφ(w~)dw~∂w¯βφ1(w¯)f~r(z~)f¯r(z¯)dν(w¯)=0, (3.26)

where (along a subsequence; see Theorem 3)

νε=d|μs|(z¯+εβw¯)|μs|((zk+1−rα1,xk+1+rα1)×⋯×(zd−rαd,xd+rαd))⇀νasε→0. (3.27)

We now take

φ(w~)=Tψ(πj(ξ~))|ξ1|β1j+⋯+|ξk|βkjρ(w~)¯,

where T_m is the Fourier multiplier operator with the symbol m. After inserting this in (3.26) and applying the Plancherel theorem with respect to w̃, we obtain:

∑r=1m∑α∈I~j,β∈I¯(−1)|α|+|β|aαβjr(z)∫Rd(2πiξ~)α|ξ1|β1j+⋯+|ξk|βkjψ(πj(ξ~))|ρ^(ξ~)|2dξ~∂w¯βφ1(w¯)f~r(z~)f¯r(z¯)dν(w¯)=0. (3.28)

If we apply here the Plancherel theorem with respect to w̄, we get

∑r=1m∑α∈I~j,β∈I¯(−1)|α|+|β|aαβjr(z)∫Rd(2πiξ~)α|ξ1|β1j+⋯+|ξk|βkjψ(πj(ξ~))|ρ^(ξ~)|2dξ~(iξ¯)βφ^1(ξ¯)f~r(z~)f¯r(z¯)ν^(ξ¯)dξ¯=0. (3.29)

From here, we conclude that if f̃(z̃) f̄(z̄) = (f̃₁(z̃) f̄₁(z̄)), …, f̃_m(z̃) f̄_m(z̄)) does not satisfy conditions of Theorem 2, we conclude that suppν̂_r = {0} which in turn implies that ν_r is the Lebesgue measure. By repeating the procedure from [3] where we replace the sphere by the manifold P̄, we conclude that the convergence from Theorem 3 used to get (3.26) is not only weak but also strong which contradicts the fact that μ_s is singular.

Let us briefly recall the arguments from [3]. First, since ψ and φ₁ in (3.29) are arbitrary, we know that it holds for every (ξ̃,ξ̄) ∈ P̃_j × ℝ^d−k, j = 1, …, n:

∑r=1m∑α∈I~j,β∈I¯(−1)|α|+|β|aαβjr(z)(2πiξ)α(iξ¯)βf~r(z~)f¯r(z¯)ν^(ξ¯)=0. (3.30)

Due to linearity, with the expense of the small right-hand side, the same holds when we replace ν by ν − ν^ε for ν^ε given by (3.27). In the matrix notation and after collecting the ξ̃-terms with the coefficients a_αβ this means

A(ξ¯)f¯(z¯)(ν^(ξ¯)−ν^ε(ξ¯))=low order terms.

We multiply the latter equation by [A(ξ̄) 𝔣̄(z̄)]^* to get

|A(ξ¯)f¯(z¯)|2(ν^(ξ¯)−ν^ε(ξ¯))=low order terms. (3.31)

If we assume that for every ξ̄ ∈ ℝ^d−k, we have

|A(ξ¯)f¯(z¯)|2≠0

and from here and (3.30) it follows that ν̂ ≡ 0 i.e. that ν is the Lebesgue measure. Moreover, we can divide (3.31) by ∣A(ξ̄) 𝔣̄(z̄)∣² from where, after invoking the Marcinkiewicz multiplier theorem and letting ε → 0, we reach to

∥ν^(ξ¯)−ν^ε(ξ¯)∥L1(K),K⊂⊂Rd.

This implies that ν^ε strongly converges toward ν which is impossible since ν is the Lebesgue measure and ν^ε are not. Thus, we conclude that for some ξ̄ it holds A(ξ̄) 𝔣̄(z̄) = 0 which we wanted to prove.

Acknowledgement

The work is supported in part by the Croatian Science Foundation under Project MiTPDE (number IP-2018-01-2449), and by the Austrian Science Fund (FWF) Lise Meitner project “Vanishing capillarity on smooth manifolds” (number M 2669-N32).

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Received: 2019-03-07

Accepted: 2020-03-09

Published Online: 2020-06-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0223

Keywords for this article

Structure of measures

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