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Normed modules, integral sequences, and integrals with variable upper limits

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Pith's one-line read The paper claims that variable-limit Lebesgue integration is categorified by a category of Banach modules, with the classical addition formula as a corollary.

desk verdict A defensible but largely definitional categorification of variable-upper-limit integrals, with two applications that do not hold up as printed; worth refereeing for the core Sections 3–4, not for the trig/global-dimension claims. read the letter →

arxiv 2411.19904 v2 pith:WW6K2NZT submitted 2024-11-29 math.CT math.CA

Miantao Liu , Yu-Zhe Liu , Shengda Liu This is my paper · ORCID
classification math.CTmath.CA MSC 16G1046B9946M40
keywords finite-dimensionalalgebrasnormedmodulescategorificationintegralswithvariableupperlimitsintegralpartiallyorderedsetsgentleglobaldimensionLebesgue-Stieltjes
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

The paper sets out to prove that integrals with variable upper limit—the map $t \mapsto \int_c^t f\,d\mu$—can be viewed as living inside the category $A^p_\Lambda$ of integral Banach modules over a finite-dimensional algebra $\Lambda$. Its central construction collects pairs $([t_1,t_2]_k, \widehat{T}_{t_2}^{t_1}(f))$ into an integral partially ordered set, defines an addition on such collections, and shows that the map that forgets the interval is a $\Lambda$-epimorphism from the image of this addition onto $\{I_\Lambda\}\times k$ (Theorem 3.14). Under explicit L-conditions the same machinery yields the classical additivity identity $\int_\alpha^\beta f+\int_\beta^\gamma f=\int_\alpha^\gamma f$ (Theorem 3.17) and recovers the known categorical description of the Lebesgue integral (Corollary 4.3). If correct, this makes the addition of integrals with different upper limits a piece of algebra rather than an analytic fact, and it gives categorical definitions of trigonometric, logarithmic, and exponential functions plus integral formulas for the global dimension of gentle algebras.

What carries the argument

The load-bearing object is the integral partially ordered set $\widehat{T}_I(f)$: for $f\in \widehat{S}_\tau(I_\Lambda)$ it is the collection of pairs $([t_1,t_2]_k,\widehat{T}_{t_2}^{t_1}(f))$ ordered by inclusion of intervals. The addition on these sets is defined by the four cases of equation (3.3), which cut overlapping or adjacent intervals and add the integrals of the restricted functions; the image of this addition carries a left $\Lambda$-module structure whose action is $(a,(S,r))\mapsto (S,\tau(a)r)$. The maps $\widehat{\int}$ and its interval-forgetting companion translate between the analytic data $(t,\int_{[\alpha,t]_\Lambda}f\,d\mu)$ and the categorical data $([\alpha,t]_k,\widehat{T}_t^\alpha(f))$. What makes the argument run is initiality: because $\widehat{S}_\tau(I_\Lambda)$ is initial, each $\widehat{T}_{t_2}^{t_1}(f)$ is the restriction of the integral over any larger interval, so adding two consecutive interval integrals in the category automatically produces the integral over the union, and the classical additivity law falls out as Theorem 3.17(1).

What would settle it

Compute both sides of Theorem 3.17(1) for $\Lambda=\mathbb{R}$, $c=0$, $d=3$, $\alpha=0$, $\beta=1$, $\gamma=2$, and $f(x)=x$ using ordinary Lebesgue integrals; if the categorical sum prescribed by equation (3.3) does not equal $([0,2],3/2)$, the additivity theorem fails. Also check whether $k=\mathbb{R}\times[0,2\pi)$ with multiplication $(r_1,\theta_1)(r_2,\theta_2)=(r_1r_2,\theta_1+\theta_2 \bmod 2\pi)$ is a field: the element $(0,\pi/2)$ has no inverse, so the trigonometric setup of Section 5.1.2 does not satisfy the paper's normed-field axioms.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the addition of variable-upper-limit integrals is controlled by the initial object of $A^p_\Lambda$. For a finite-dimensional algebra $\Lambda$ with basis $b_i$ and an interval $I=[c,d]_k$, the completed space $\widehat{S}_\tau(I_\Lambda)$ of elementary simple functions on $I_\Lambda=\sum_i [c,d]_k b_i$ is, by a theorem quoted from [21], initial in $A^p_\Lambda$; every object therefore receives exactly one morphism $\widehat{T}_{(N,v,\delta)}$. The paper's new move is to take the family $\{([t_1,t_2]_k,\widehat{T}_{t_2}^{t_1}(f))\}$ as an integral partially ordered set, to define an addition on these families by gluing intervals and adding the integrals of the corresponding indicator functions, and to prove that the image of this addition is a left $\Lambda$-module. Theorem 3.14 states that the interval-forgetting restriction to this image is a $\Lambda$-epimorphism onto $\{I_\Lambda\}\times k$, which is the categorical formulation of adding integrals with different upper limits. Theorem 3.17 then shows that under the L-conditions the additivity law holds for the corresponding numbers, and Corollary 4.3 identifies the i-poset space with the space of continuous functions vanishing at $c$, thereby recovering the known categorical derivation of the Lebesgue integral and showing the new object represents absolutely continuous functions.

Load-bearing premise

The construction rests on previously established theorems, quoted here without proof, that the completed step-function space is initial in $A^p_\Lambda$ and that its unique morphism to the scalar field is the Lebesgue integral; if either theorem is unsound, the epimorphism, the additivity law, and Corollary 4.3 all collapse.

Editorial extensions

If this is right

  • The classical additivity $\int_\alpha^\beta f\,d\mu + \int_\beta^\gamma f\,d\mu = \int_\alpha^\gamma f\,d\mu$ is a direct corollary of the categorical addition structure (Theorem 3.17(1)).
  • The space of i-posets is R-linearly isomorphic to the Banach space of continuous functions on $[c,d]$ vanishing at $c$, recovering the known categorical derivation of the Lebesgue integral and giving a categorical model of absolutely continuous functions (Corollary 4.3).
  • Trigonometric functions arise as inverses of integrals with variable upper limit, and their period $2\pi$ is computed from the circumference of a circle via a curvilinear integral in the same framework.
  • Logarithmic and exponential functions arise from the i-poset of $1/t$: $\ln y_1+\ln y_2=\ln(y_1y_2)$ and $e^{x_1}e^{x_2}=e^{x_1+x_2}$ follow from the addition rule.
  • For gentle algebras with finite global dimension, $\mathrm{gl.dim}\,A/2$ is a supremum of multiple Lebesgue integrals over forbidden threads of the algebra and over permitted threads of its Koszul dual, and $\mathrm{gl.dim}\,A$ is a supremum of Lebesgue–Stieltjes integrals of the projections of permitted threads (Theorems 5.5 and 5.6).

Reading between the lines

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

  • The same initial-object-plus-addition recipe would work for any integral transform whose kernel defines a morphism from the initial step-function object, so a categorical addition formula for Laplace or Fourier transforms is a plausible next step, though the paper does not state this.
  • The trigonometric construction suggests a categorical route to elliptic functions: taking inverses of elliptic integrals with variable upper limits inside $A^p_\Lambda$ could produce functions whose double periodicity and addition theorems might be derived from the same i-poset addition; this is an extension, not a claim of the paper.
  • The global-dimension formulas express a homological invariant as an integral of vertex homomorphisms; if the quoted equality $\mathrm{gl.dim}\,A=\sup_F \ell(F)$ holds for a wider class of quadratic monomial algebras, the identity $\ell(q)=2\int v_q$ would give integral characterizations beyond gentle algebras.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 6 minor

Summary. The paper develops a categorical framework for integrals with variable upper limits in the spirit of Leinster's A^p categories, building on the authors' prior work on normed modules over finite-dimensional algebras. It introduces integral partially ordered sets bTI(f), an addition rule (3.3) on pairs of the form ([u,v], bT_v^u(f)), and maps ⅁, ⅁♮ relating these pairs to elements of Σ(IΛ)×k. The central results are Theorem 3.14 (the restriction of ⅁♮ to Im(+) is a Λ-epimorphism), Theorem 3.17 (additivity and k-linearity of integrals with variable upper limits under L-conditions), and Corollary 4.3, which purports to recover the Leinster-Meckes categorification of the Lebesgue integral with variable upper limit. Two applications are offered: a categorified account of elementary functions (trigonometric, logarithmic, exponential) and new formulas for the global dimension of gentle algebras in terms of multiple and Lebesgue-Stieltjes integrals.

Significance. If the main theorems were fully established, the paper would contribute a novel categorical language for the map t ↦ ∫_α^t f dµ and its additivity, connecting Leinster's categorification to variable upper limits. The authors are systematic and careful in verifying the k-linear and Λ-module structures on the images of + and ⅁ (Lemmas 3.5, 3.11), and they clearly state their dependence on the companion papers [20,21]. However, as documented below, several load-bearing points are not proved as written, and one application uses an invalid base field. The genuinely new content beyond the quoted results of [20,21] and the definitions is currently limited, and the advertised scope of the main theorems needs substantial revision.

major comments (5)
  1. [§3.3, Lemma 3.8 and Theorem 3.14] The proof of Lemma 3.8 defines f = k/µ(S) · 1_S and asserts bT_{k,µ(IΛ),m}(f) = k, which requires the identity bT(1_S) = µ(S) for arbitrary measurable S. Theorem 2.17(1) only states bT(1_{IΛ}) = µ(IΛ), k-linearity, and the inequality |bT(|f|)| ≤ |bT(f)|; it does not determine bT on arbitrary measurable subsets. This equality is known for the Lebesgue integral under the L-conditions, but Lemma 3.8 is stated before L-conditions are imposed and Theorem 3.14 is claimed for all A^p. Moreover, the construction is undefined when µ(S)=0. Consequently, the surjectivity of ⅁♮⅁, and hence the Λ-epimorphism assertion of Theorem 3.14, is not established for the general framework.
  2. [§3.3, proof of Theorem 3.14] The proof begins: 'by Proposition 3.13, we have known that Im(+) is a left Λ-submodule of Im(⅁)'. This reverses the actual statement of Proposition 3.13, which says Im(⅁) is a left Λ-submodule of Im(+). The proof then verifies Λ-linearity but does not prove surjectivity; the final sentence 'Since dim_k({IΛ}×k)=1, we find that ⅁♮|Im(+) is a Λ-epimorphism' is a non sequitur unless one already knows the map is nonzero or surjective, which would require an appeal to Lemma 3.8 (whose proof is incomplete, as noted above).
  3. [§3.4, Theorem 3.17(1)] The additivity identity in Theorem 3.17(1) is a direct consequence of the definition of the addition + in (3.3). The proof invokes Lemma 3.4, and that lemma is proved immediately by unpacking the cases of (3.3). Thus the identity holds by construction rather than being a substantive consequence of the categorical structure. The paper should either present this as a consistency check of the definition of + or provide an independent argument showing why the definition is forced by the categorical framework; the current framing overstates the content.
  4. [§5.1.2, conditions (2')–(6')] The base object k = Λ = R×[0,2π) with multiplication (r1,θ1)·(r2,θ2)=(r1r2,θ1+θ2 mod 2π) is not a normed field containing R. For any θ>0, (0,θ)·(1,2π−θ)=(0,0), so nonzero zero divisors exist, and no field addition is specified on R×[0,2π). Since Theorem 2.17 and the definition of A^p require k to be a complete field, the derivation of K=π/2 and the identification of the functions s and c with sine and cosine are not justified.
  5. [§5.4–5.5, Theorems 5.5 and 5.6] The global-dimension formulas are, in substance, restatements of the authors' earlier result [20, Theorem 5.10] (quoted in (5.8)) together with the elementary computation (5.7), which essentially rewrites the length ℓ(q) as 2∫v_q dµ. The Stieltjes-integral formula in Theorem 5.6 is not rigorously derived: φ_l in (5.13) is written as a function x↦ln x^l rather than as a measure, and the chain of equalities leading to (L-S)∫_1^2 w_P|kP dφ_l = l omits the measure-substitution hypotheses. The claims of a 'new approach' for characterizing global dimensions should be toned down, and the derivations should be completed.
minor comments (6)
  1. [General notation] The symbol ⅁ is used for several different maps (⅁, ⅁♮, ⅁α, ⅁α,f) with closely related meanings; more mnemonic or indexed notation would greatly improve readability.
  2. [§2, Definition 2.7] The definition of elementary simple functions writes f(k1,...,kn)=Σ k_i 1_{X_i}, but the domain IΛ is a subset of Λ; the meaning of the coordinates k_i and the sets X_i should be clarified.
  3. [§3.3.1, equation (3.5)] The definition of ⅁α uses the condition α ∈ (c,d)k, but the set Sα,f is then described for α ≤ t ≤ d, which includes the case α=c; please make the domain consistent.
  4. [§3.3.1, Lemma 3.10] The statement '⅁ is an injection' is made for the map on the disjoint union S over all α and f; the proof only considers equality of images of two elements, which is fine, but the domain of ⅁ should be stated explicitly to avoid confusion with the restriction ⅁α.
  5. [§5.3.2, computation of bTI(v_q)] The computation contains the expression ∫_0^t k_{e_v} d k_{e_v} = t^2/2, which is nonstandard notation; it should be rewritten as an ordinary integral over a real variable with a clear substitution.
  6. [Throughout] There are numerous typographical errors and infelicities (e.g., 'defied' for 'defined', 'structrue' for 'structure', inconsistent use of k and R). The manuscript would benefit from a careful proofreading pass.

Circularity Check

3 steps flagged · score 6.0 of 10

The central additivity theorem is encoded in Eq. (3.3) by construction; the global-dimension applications reduce to the authors' own [20, Theorem 5.10], and Lemma 3.8 silently assumes a stronger measure-recovery property than Theorem 2.17 provides.

  1. self definitional [Section 3.2, Eq. (3.3) and Lemma 3.4; used in Theorem 3.17(1) and Theorem 1.2]
    "The addition functor + is defined by ([u, v]k, bT_v^u(f)) + ([s, t]k, bT_t^s(g)) := (U, bT_{min{u,v,s,t}}^{max{u,v,s,t}}(1_{[u,v]_Λ} f + 1_{[s,t]_Λ} g)) (3.2); ... If u ⪯ v = s ⪯ t, then U = [u, t]k, and ... ([u, v]k, bT_v^u(f)) + ([s, t]k, bT_t^s(f)) = ([u, t]k, bT_t^u(f)). Lemma 3.4"

    The branch ♠ of Eq. (3.3) with u ⪯ v = s ⪯ t is precisely the interval-telescoping identity; Lemma 3.4 states it as a “direct corollary” of (3.3). Theorem 3.17(1) applies ⅁♮ to this identity and presents the result as the additivity of integrals with variable upper limits. The categorified additivity is therefore loaded into the definition of the + operation; the theorem unpacks the definition rather than deriving the additivity from the categorical structure. This is pattern 1: the claimed prediction is equivalent to its input by construction.

  2. other [Section 3.3.2, Lemma 3.8 proof; inherited by Theorem 3.14 and Theorem 1.1]
    "For any pair (S, k) in Σ(IΛ) × k, let f = k/µ(S) · 1S : IΛ → k, ... then we obtain bT_(k,µ(IΛ),m)(f) = (A^p) ∫_{IΛ} f dµ = k/µ(S) · (A^p) ∫_{IΛ} 1_S dµ = k/µ(S) · µ(S) = k."

    The equality (A^p)∫_{IΛ} 1_S dµ = µ(S) for arbitrary measurable S is not among the properties of bT quoted in Theorem 2.17(1), which gives only bT(1_{IΛ}) = µ(IΛ), k-linearity, and an inequality. Lemma 3.8 silently assumes the measure-recovery property that the categorified integral is supposed to supply, so the surjectivity proof of ⅁♮∘⅁, and hence Theorem 3.14's Λ-epimorphism, is not established for general A^p; it is valid only under the L-conditions, where bT is already known to be Lebesgue integration. This is an unsupported load-bearing assumption rather than an independent derivation.

1 more flagged steps
  1. self citation load bearing [Section 5.4.2, Eq. (5.8)–(5.10), Theorems 5.5 and 5.6]
    "From [20, Theorem 5.10], we have the following fact gl.dimA = sup_{F ∈ forb(A)} ℓ(F). (5.8) Moreover, one can check the correspondence forb(A) → perm(A!), a_1···a_l ↦ a_l^!···a_1^!, is bijective. Then, by (5.8) and (5.7), we have gl.dimA = sup_{F ∈ forb(A)} ℓ(F) = sup_{F ∈ forb(A)} (A^1_A) ∫_{[0,1]_A} 2v_F dµ, (5.9)"

    Theorems 5.5 and 5.6 are obtained by substituting the elementary identity ℓ(q) = 2∫ v_q dµ (Eq. 5.7) into the formula gl.dim A = sup_{F ∈ forb(A)} ℓ(F), and that formula is imported verbatim from [20, Theorem 5.10], a prior paper by the same author group. No proof of the global-dimension formula appears here, so the advertised 'new approach' to global dimensions reduces to a load-bearing self-citation plus a computation; the central content is not independently derived in this paper.

full rationale

The paper's own additivity theorem for integrals with variable upper limits, Theorem 1.2(1) = Theorem 3.17(1), is true by construction: the binary operation + in Eq. (3.3) is defined so that adjacent intervals telescope, Lemma 3.4 records this as a direct corollary, and Theorem 3.17(1) only applies ⅁♮ to that definitional identity. This is the clearest circular step and is reflected in score 6, not higher, because the construction of the categories A^p_Λ, the Λ-module structure on Im(+), and the Leinster-Meckes recovery of Corollary 4.3 (quoted from the external [17]) retain independent mathematical content. In addition, Lemma 3.8's surjectivity proof assumes bT(1_S)=µ(S) for arbitrary measurable S, a property not stated in Theorem 2.17(1); this is a missing-support gap in the proof of Theorem 3.14, though it is repaired under the L-conditions where bT is already Lebesgue integration. Finally, the global-dimension applications in Section 5 rest almost entirely on the authors' own [20, Theorem 5.10]; the new integral formulas are a repackaging of that cited theorem. The proposed field k = R×[0,2π) with angle-pointwise multiplication is not actually a field, but that is a correctness risk rather than a circularity and does not affect this score.

Assumptions & free parameters 2 free parameters · 8 assumptions · 3 invented entities

The central claim is carried by five imported assumptions: the initial-object and integration theorems of [21] (Theorems 2.16 and 2.17), the global-dimension characterization of [20, Theorem 5.10], the change-of-variables facts of [13], and the circumferential definition of π. On top of these, the trigonometric application adds an ad hoc object, R × [0, 2π), that is treated as a field but is not one. The genuinely independent residue, equation (5.7) and the module lemmas, is computational.

free parameters (2)
  • K = π/2 (claimed)
    K = bT_0^1(f) = ∫_0^1 (1-t²)^{-1/2} dt is the period constant of the functions s and c in §5.1. The paper claims 2K = π, but the derivation imports π through the circumference formula and contains a factor-of-two error, so the value is asserted rather than derived.
  • φ_l (Stieltjes measure family) = φ_l(x) = ln(x^l)
    The Stieltjes measures in Theorem 5.6 are chosen so that (L-S)∫_1^2 w_P dφ_l = l = ℓ(P): the integral of the identity function against d(ln x^l) equals l by construction, matching the known path-length value from [20, Theorem 5.10].
assumptions (8)
  • domain assumption The object (Ŝτ(IΛ), 1_{IΛ}, γ̂ξ) is an initial object of A^p_Λ (Theorem 2.16, quoted from [21, Theorem 6.3]).
    Invoked throughout (Definition 3.2, Corollary 4.3(2)); the existence of the integral morphisms bT over every subinterval is assumed from this overlapping-author preprint and not re-proved.
  • domain assumption The unique morphism bT_{(k,µ(IΛ),m)} satisfies bT(1_S) = µ(S), is a homomorphism of k-modules, and equals the Lebesgue integral when the L-conditions hold (Theorem 2.17, quoted from [21, Theorems 7.6] and [17]).
    Load-bearing for Lemma 3.8's surjectivity construction, Theorem 3.14, Theorem 3.17, and Corollary 4.3.
  • domain assumption gl.dim A = sup_{F ∈ forb(A)} ℓ(F) for gentle algebras with finite global dimension ([20, Theorem 5.10]).
    The actual content of Theorems 5.5 and 5.6; the paper translates this self-authored published theorem into integral formulas rather than deriving a new characterization.
  • domain assumption The circumference of a Euclidean circle of radius R equals 2πR.
    Used in §5.1.2 to conclude K = π/2; this is the defining property of π, so the claimed derivation of 2K = π assumes the value it purports to prove.
  • ad hoc to paper k = R × [0, 2π) with the polar-coordinate multiplication is a normed field.
    Assumed in conditions (1')-(6') of §5.1.2; false as stated because no addition is defined, (0,θ) represents zero non-uniquely, and the L-condition (L2) requires k = R or k = C.
  • domain assumption Change-of-variables and Lebesgue-Stieltjes transference theorems for functions defined on algebras ([13, Theorem 3.5, Corollary 3.6, Lemma 3.4]).
    Used for the logarithm section, equation (5.3), and equations (5.11)-(5.12) in the Stieltjes section; imported from an overlapping-author preprint.
  • domain assumption I satisfies the segmentation condition: every subinterval [c', d'] of I splits into two order-isomorphic halves via κ_{c'} and κ_{d'}.
    Assumed after Definition 3.1; it makes the integral posets bTI(f) definable over every subinterval. It holds for real intervals with affine maps.
  • standard math The space Ŝτ(IΛ) of equivalence classes of elementary simple functions completes to L1([c,d]) when Λ = R (Leinster's identification, Remark 2.12).
    Background for Corollary 4.3(2) and Remark 4.4; standard completion of simple functions.
invented entities (3)
  • Integral partially ordered sets bTI(f)
    purpose: Bookkeeping for the family of pairs ([t1, t2], integral of f over [t1, t2]); the paper's candidate categorification of integrals with variable upper limits.
    An internal construction whose content presupposes the integral from [21]; Remark 4.4(3) claims it categorifies absolute continuity, but that property is classical for indefinite Lebesgue integrals.
  • The object k = R × [0, 2π) used as base field
    purpose: To carry the circumference computation in §5.1.2 and thereby identify the period constant K with π/2.
    Not a field: addition is never defined, and (0, θ) is a non-unique zero; the category A^1_{R×[0,2π)} is not an instance of the paper's framework.
  • Lebesgue-Stieltjes measure family φ_l with φ_l(x) = ln(x^l)
    purpose: To make (L-S)∫_1^2 w_P|_{kP} dφ_l equal the path length ℓ(P) in Theorem 5.6.
    The measure is engineered so the integral of the identity function equals l; the resulting formula re-encodes the known length-based characterization of [20] instead of measuring gl.dim independently.

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Pith. "Pith review of Normed modules, integral sequences, and integrals with variable upper limits." pith.science (2026). https://pith.science/paper/WW6K2NZT

@misc{pith2026241119904,
  author       = {Pith},
  title        = {Pith review of: Normed modules, integral sequences, and integrals with variable upper limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WW6K2NZT}},
  note         = {Machine review of arXiv:2411.19904}
}
abstract

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathit{\Lambda}$ and the category $\mathscr{A}^p_{\mathit{\Lambda}}$ associated with $\mathit{\Lambda}$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.

Figures

Figures reproduced from arXiv: 2411.19904 by the authors.

Figure 2.1
Figure 2.1. Order-preserving bijections κc and κd Definition 2.6. We define step function spaces Eu on IΛ as the set of all functions of the form X (xiyi)i k(xiyi)i1(xiyi)i (= X i ki1Ii for simplification), where k(xiyi)i ∈ k, and (xiyi)i corresponds to the Cartesian product Qn i=1 I (xiyi) . Eu is a left Λ-module defined as the pair (Eu, hEu : Λ → EndkEu), where hEu is given by hEu : a 7→ [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗
Figure 2.2
Figure 2.2. Juxtaposition map (in the case for dimk Λ = 2) (L2) k = R = Λ (in this case, the norms of k and Λ coincide), or k = C = Λ (in this case, the norm of C is defined by the modulus of complex numbers); (L3) I = [x1, x2] (= IR); (L4) ξ = x1 + x2 2 , κx1 (x) = x + x1 2 , κx2 (x) = x + x2 2 ; (L5) τ = idk; (L6) µ : Σ(IΛ) → R ≥0 is the Lebesgue measure defined on the σ-algebra Σ(IΛ), then there exists an object of the form … view at source ↗

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