Pith. sign in

REVIEW 4 major objections 4 minor 44 references

On Lebesgue points and measurability with Choquet integrals

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a nonnegative function with locally finite Choquet integrals is Lebesgue measurable exactly when it is the almost-everywhere limsup of its dyadic Hausdorff ball averages.

desk verdict Worthwhile paper with a new measurability characterization and sharp counterexamples; the proof of Proposition 5.2 has a reparable gap and the paper leans on an unpublished manuscript, but the core results stand up to a referee. read the letter →

arxiv 2502.03267 v2 pith:B7WS5M7L submitted 2025-02-05 math.FA

Petteri Harjulehto , Ritva Hurri-Syrjänen This is my paper · ORCID
classification math.FA MSC 28A2528A2042B25
keywords ChoquetintegraldyadicHausdorffcontentcapacityLebesguepointmaximaloperatornon-measurablefunctionquasicontinuitymeasurability
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 studies Lebesgue points for Choquet integrals taken with respect to the dyadic Hausdorff content, a set function that assigns a size to every set, measurable or not. Its central result characterizes Lebesgue measurability: for a non-negative function with finite Choquet integrals over balls, the function is Lebesgue measurable if and only if $f(x)=\limsup_{r\to 0+}\frac{1}{\widetilde H^n_\infty(B(x,r))}\int_{B(x,r)} f\,d\widetilde H^n_\infty$ holds for $\widetilde H^n_\infty$-almost every $x$. The forward direction is a Lebesgue-point theorem for measurable functions, while the reverse direction shows that the limsup of Choquet ball averages is itself measurable, so a function recovered from its averages cannot conceal non-measurability. The paper also proves convergence theorems, weak-type maximal estimates, and a density result showing that continuous functions are dense in the $n$-dimensional space but not in the spaces of dimension $\delta

What carries the argument

The central objects are the $\delta$-dimensional dyadic Hausdorff content $\widetilde H^\delta_\infty$, defined by covering sets with countably many dyadic cubes and minimizing the sum of side-length powers, and the Choquet integral $\int_\Omega f\,d\widetilde H^\delta_\infty=\int_0^\infty \widetilde H^\delta_\infty(\{f>t\})\,dt$. The operative mechanism is the ball-average map $f^\delta_{B(x,r)}$ and its limsup as $r\to0+$. The argument hinges on showing that $x\mapsto f^\delta_{B(x,r)}$ is lower semicontinuous, that the resulting limsup is Lebesgue measurable, that the $n$-dimensional maximal operator satisfies a weak-type estimate, and that being an $nL^1$-limit of continuous functions is equivalent to $\widetilde H^\delta_\infty$-quasicontinuity.

What would settle it

Compute $\widetilde H^n_\infty(B(x,r))$ and $\widetilde H^n_\infty(B(y,r))$ for two balls of equal radius whose centers differ by a non-dyadic vector, using the dyadic cube cover definition; unequal values would disprove the equality used in Proposition 5.2(1), while equal values for every such pair would validate the lower semicontinuity step and with it the reverse direction of Corollary 1.1.

Watch

Extended reading notes

Core claim

The paper's central discovery is the equivalence in Corollary 1.1. Given $f:\mathbb R^n\to[0,\infty]$ with $\int_B f\,d\widetilde H^n_\infty<\infty$ for every open ball $B$, the function $f$ is Lebesgue measurable exactly when $f(x)=\limsup_{r\to0+} f^n_{B(x,r)}$ for $\widetilde H^n_\infty$-almost every $x$, where $f^n_{B(x,r)}$ denotes the Choquet average of $f$ over $B(x,r)$ with respect to the $n$-dimensional dyadic Hausdorff content. The forward direction runs through the Lebesgue-point theorem for functions that are $nL^1$-limits of continuous functions, using the fact that at $\delta=n$ the Choquet integral is comparable to the Lebesgue integral and that continuous functions are dense in $L^1$. The reverse direction uses lower semicontinuity of the ball-average map and measurability of its limsup to conclude that any function satisfying the recovery formula agrees almost everywhere with a Lebesgue measurable function.

Load-bearing premise

The argument that the limsup of ball averages is measurable depends on the claim that the dyadic Hausdorff content of a ball does not change when the ball's center is moved, so that the map $x\mapsto f^\delta_{B(x,r)}$ is lower semicontinuous; if that translation invariance fails, the measurability of the limsup needs a different proof.

Editorial extensions

If this is right

  • A Lebesgue measurable non-negative function with locally finite Choquet integral is recovered, up to an $\widetilde H^n_\infty$-null set, by the limsup of its dyadic-Hausdorff ball averages.
  • The same recovery formula forces measurability: functions that are not Lebesgue measurable cannot satisfy the average-recovery identity almost everywhere.
  • For $\delta=n$, every Lebesgue measurable $f\in nL^1(\mathbb R^n,\widetilde H^n_\infty)$ is an $nL^1$-limit of continuous functions, so continuous functions are dense in that space.
  • For $0<\delta<n$, Lebesgue measurable functions such as characteristic functions of balls or snowflake-type domains fail the recovery formula at boundary points, showing that continuous functions are not dense in $nL^1(\mathbb R^n,\widetilde H^\delta_\infty)$.
  • Being an $nL^1$-limit of continuous functions is equivalent to $\widetilde H^\delta_\infty$-quasicontinuity, and such functions have Choquet Lebesgue points almost everywhere.

Reading between the lines

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

  • Editorial inference: if the reverse-direction proof can be completed without exact translation invariance of ball content, the characterization may extend to other capacities whose ball masses are merely comparable, since only lower semicontinuity of the average map and a weak-type maximal estimate are needed.
  • Editorial inference: the examples suggest a dimension-threshold phenomenon, namely that failure of Lebesgue-point recovery is tied to boundary sets of positive $\delta$-dimensional content; one could test whether centering dyadic cubes at $x$ instead of balls restores recovery for all $0<\delta\le n$.
  • Editorial inference: the characterization offers a route to detect non-measurability numerically by approximating the limsup of Choquet averages on a grid and comparing it with the function on the complement of a small exceptional set.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper develops Lebesgue point theory for Choquet integrals with respect to dyadic Hausdorff content, working with functions that are not assumed Lebesgue measurable. It proves auxiliary monotone convergence and Fatou-type results, maximal-function estimates, a norm convergence theorem, density results for continuous functions in the spaces nL1(R^n, H^δ∞), and a central characterization (Corollary 1.1): for nonnegative f with finite Choquet integrals on every open ball relative to H^n∞, f is Lebesgue measurable if and only if f equals the limsup of its Choquet averages on balls, H^n∞-almost everywhere. The paper also gives two examples for δ<n showing that Lebesgue measurable functions need not be nL1-limits of continuous functions.

Significance. If Corollary 1.1 is correct, it is a genuinely interesting result: it characterizes Lebesgue measurability purely through the pointwise behavior of Choquet averages with respect to dyadic Hausdorff content. The companion density and non-density results for continuous functions in nL1(R^n, H^δ∞), especially the counterexamples for δ<n, are also valuable and are supported by explicit constructions. The paper builds on published work [7] for the weak-type maximal estimate, and it gives self-contained convergence and approximation arguments for much of the development. However, the proof of the key measurability assertion in Proposition 5.2 contains a real gap, and several auxiliary results are imported from the authors' own unpublished submitted manuscript [21], which makes the current version difficult to verify as submitted.

major comments (4)
  1. [Section 5, Proposition 5.2(2)] The proof of claim (2) says that limsup_{r→0+} f^δ_{B(x,r)} is Lebesgue measurable 'as a pointwise limit of Lebesgue measurable functions.' This is not justified: the limsup is over the uncountable set r>0 and is not a sequential pointwise limit. The measurability of this limsup is exactly what Proposition 5.3 and the reverse direction of Corollary 1.1 require, so the gap is load-bearing. The claim is repairable: if part (1) is available, then G_k(x)=sup_{0<r<1/k} f^δ_{B(x,r)} is lower semicontinuous as a supremum of lower semicontinuous functions, and L(x)=inf_k G_k(x) is Borel measurable. The authors should replace the 'clear' argument with this argument or an equivalent one.
  2. [Section 5, Proposition 5.2(1)] The proof of lower semicontinuity uses the equality H^δ∞(B(x,r)) = H^δ∞(B(y,r)) for balls with different centers. Dyadic Hausdorff content with respect to a fixed dyadic lattice is not translation invariant when δ<n, so this equality is not available in the stated generality. For δ=n the equality is valid because the dyadic n-dimensional content of an open Euclidean ball agrees with its Lebesgue measure, hence is translation invariant, but for δ<n the proof of Proposition 5.2(1) is incomplete as written. If Proposition 5.2 is needed only for δ=n in the proof of Corollary 1.1, the authors should say so and justify the δ=n equality separately; otherwise a different argument is needed for δ<n.
  3. [Sections 3 and 4, Theorem 4.11, Remark 3.3] The paper repeatedly relies on the authors' unpublished submitted manuscript [21] for load-bearing results: Remark 3.3 (comparability of the Choquet integral with the Lebesgue integral), Theorem 4.11 (strong-type estimate for Mn, proved in one line from [21, Theorem 4.6]), the ball-content comparability H^δ∞(B(x,r))≈r^δ used in Examples 6.1 and 6.2, and the definition and properties of nL1(R^n, H^δ∞). Since [21] is not available to referees or readers, these results cannot be checked from the submitted manuscript. The authors should include the relevant statements and proofs from [21], or replace them with published references, so that the central claims of the paper are self-contained in the respects on which they depend.
  4. [Section 2, definition of H^δ∞] The covering condition 'E ⊂ int(∪ Qi)' is problematic for the half-open dyadic cubes defined immediately before it. For example, the origin is never contained in the interior of a dyadic cube of the form [m2^k,(m+1)2^k)^n in the fixed lattice, so under the literal definition H^δ∞ of any set containing the origin would be infinite. This would make the assumption in Corollary 1.1 fail for very simple functions such as f≡1 on a ball containing the origin. The authors should clarify the intended convention, presumably E⊂∪ Qi or E⊂∪ int(Qi) as in [42], and ensure that the properties (H1)-(H5) are stated for that convention.
minor comments (4)
  1. [Example 6.2] In the sentence 'χΩ(x) /nequallim sup', the symbol '/nequal' should be '≠'.
  2. [Remark 4.10] There is a typo: 'resent results' should be 'recent results'.
  3. [Throughout] There are several typographical inconsistencies, including 'continuos' in the abstract, 'difference' and 'different' with nonstandard ligature breaks, and the garbled phrase 'f (x)] d ˜H δ∞' in the proof of Theorem 4.11. These should be corrected in a final revision.
  4. [Section 5, proof of Proposition 5.2(1)] In the chain of inequalities after the choice of η, the ratio H^δ∞(B(x,r))/H^δ∞(B(y,r)) is written as a multiplier; once the equality of contents is removed (as it must be for δ<n), the displayed calculation should be rewritten in terms of a comparability constant, and the openness of the superlevel set should be verified with that constant.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main equivalence is proved from external maximal estimates and self-contained convergence arguments; the self-citations to [21] are supporting rather than definitional, and the main caveats are proof gaps, not circularity.

full rationale

The claimed equivalence in Corollary 1.1 is not derived from itself. The forward direction runs through Corollary 5.17 and Theorem 5.12, whose only external input is the weak-type maximal estimate of [7, Theorem A] (Theorem 4.8), together with the standard density of continuous functions in L1; the convergence lemmas (Propositions 4.1, 4.2, Theorem 4.6) are proved in the paper. The reverse direction reduces to Lebesgue measurability of the limsup of x↦f^n_{B(x,r)}, supplied by Proposition 5.2. Self-citations to the authors' submitted [21] occur for the standard comparison (3.4) used in Theorem 5.9(1), for the statement of the nL1 spaces and maximal operator, and for ball-content comparability in the counterexamples; none of these defines the target equivalence. In particular, the strong-type estimate [21, Theorem 4.6] is used only in the auxiliary Theorem 4.11, not in the proof of Corollary 1.1. Two caveats should be flagged, but they are correctness issues rather than circularity. First, Proposition 5.2(2) says the uncountable limsup is Lebesgue measurable 'as a pointwise limit of Lebesgue measurable functions'; this is not a valid sequential statement, though it is repairable via lower semicontinuity and a k-limit. Second, Proposition 5.2(1) uses H^δ∞(B(x,r)) = H^δ∞(B(y,r)), which is false for a fixed dyadic grid when δ<n, but the central corollary only needs δ=n where translation invariance holds. Overall, no prediction or characterization is reduced by construction to fitted inputs; the central claim has independent content. Score 2 only because a few supporting comparability facts are cited from the same authors' unpublished [21] rather than proved or cited to a published source.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central arguments use standard measure-theoretic results and the stated properties of dyadic Hausdorff content. The paper adds no free parameters or invented entities. Its main nonstandard inputs are properties from the authors' own unpublished [21] and a geometric assertion about the von Koch snowflake, both listed as axioms and domain assumptions.

assumptions (5)
  • domain assumption The dyadic Hausdorff content H^delta_infty satisfies properties (H1)-(H5) and strong subadditivity (2.1).
    Stated in Section 2 and taken from [42, Theorems 2.1, Propositions 2.3, 2.4].
  • domain assumption The comparability H^delta_infty(B(x,r)) approximately r^delta holds with constants depending only on n.
    Used throughout, e.g., in Proposition 5.2 proof, Example 6.1, and Theorem 4.11; cited to [21, (2.3) and Proposition 2.5] and [42, Proposition 2.3].
  • domain assumption The strong-type estimate for the centred maximal operator Mn on nL1 spaces, i.e., ||Mn f||_{nL1} <= c ||f||_{nL1}, holds.
    Used in the proof of Theorem 4.11, cited to the authors' unpublished manuscript [21, Theorem 4.6].
  • domain assumption Every H^delta_infty-quasicontinuous function is an nL1-limit of continuous functions.
    Used in the proof of Theorem 5.11, cited to [33, Proposition 3.2].
  • domain assumption For the modified von Koch snowflake in Example 6.2, there exists c in (0,1) such that every x on the boundary and every small r has a ball B(y,cr) contained in Omega intersect B(x,r).
    This geometric property is asserted without proof in Example 6.2, relying on the construction from [22, Section 3].

how reviews work

0 comments
Cite this review

Pith. "Pith review of On Lebesgue points and measurability with Choquet integrals." pith.science (2026). https://pith.science/paper/B7WS5M7L

@misc{pith2026250203267,
  author       = {Pith},
  title        = {Pith review of: On Lebesgue points and measurability with Choquet integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7WS5M7L}},
  note         = {Machine review of arXiv:2502.03267}
}
read the original abstract

We consider Choquet integrals with respect to dyadic Hausdorff content of non-negative functions which are not necessarily Lebesgue measurable. We study the theory of Lebesgue points. The studies yield convergence results and also a density result between function spaces. We provide examples which show sharpness of the main convergence theorem. These examples give additional information about the convergence in the norm also, namely the difference of the functions in this setting and continuous functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 42 canonical work pages

  1. [7]

    B., Roychowdhury, P

    Basak, R., Chen, Y .-W . B., Roychowdhury, P . and Spector,D.: The capacitary John– Nirenberg inequality revisited, https://arxiv.org/abs/2501.11412 18 PETTERI HARJULEHTO AND RITV A HURRI-SYRJ ¨ANEN

    arXiv

  2. [21]

    Harjulehto, P ., Hurri-Syrj¨ anen, R.: On Hausdor ffcontent maximal operator and Riesz potential for non-measurable functions, submitted m anuscript

    work page

  3. [42]

    Y ang D., Y uan, W .: A note on dyadic Hausdor ffcapacities, Bull. Sci. Math. 132 (2008), 500–509

    work page 2008

  4. [1]

    R.: A note on Riesz potentials, Duke Math

    Adams, D. R.: A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778

    work page 1975

  5. [2]

    R.: A note on Choquet integrals with respect to H ausdorffcapacity

    Adams, D. R.: A note on Choquet integrals with respect to H ausdorffcapacity. In: Cwikel, M., Peetre, J., Sagher, Y ., Wallin, H. (eds.) Functi on Spaces and Applica- tions (Lund 1986), Lecture Notes in Mathematics vol. 1302, S pringer, Berlin (1988) pp. 115–124

    work page 1988

  6. [3]

    R.: Choquet Integrals in Potential Theory, Publ

    Adams, D. R.: Choquet Integrals in Potential Theory, Publ. Mat. 42 (1998), 3–66

    work page 1998

  7. [4]

    R.: Morrey Spaces, Birkh¨ auser, Cham–Heidelb erg–New Y ork, 2015

    Adams, D. R.: Morrey Spaces, Birkh¨ auser, Cham–Heidelb erg–New Y ork, 2015

    work page 2015

  8. [5]

    R., Xiao, J.: Bochner–Riesz means of Morrey fun ctions, J

    Adams, D. R., Xiao, J.: Bochner–Riesz means of Morrey fun ctions, J. F ourier Anal. Appl. 26 (2020), no. 1, Paper No. 7

    work page 2020

Show all 44 references
  1. [6]

    A.: Sobolev Spaces, Academic Press, Inc., Lond on, 1975

    Adams, R. A.: Sobolev Spaces, Academic Press, Inc., Lond on, 1975

    1975

  2. [8]

    Cerd´ a, J., Mart´ ın, J., Silvestre, P .: Capacitary func tion spaces. Collect. Math. 62 (2011), no. 1, 95–118

    2011

  3. [9]

    Chen, Y .-W ., Ooi, K.H., Spector, D.: Capacitary maximal inequalities and applica- tions. J. Funct. Anal. 286 (2024), issue 12, paper no. 110396

    2024

  4. [10]

    Chen, Y .-W ., Spector, D.: On functions of bounded β-dimensional mean oscillation, Adv. Calc. V ar .17 (2024), no. 3, 273-296

    2024

  5. [11]

    Choquet, G.: Theory of capacities. Ann. Inst. F ourier (Grenoble) 5 (1953–1954), 13–295

    1953

  6. [12]

    Dafni, G., Xiao, J.: Some new tent spaces and duality the orems for fractional Car- leson measures and Qα(/bbRn), J. Funct. Anal. 208 (2004), 377–422

    2004

  7. [13]

    27, Klu wer Academic Publish- ers Group, Dordrecht, 1994

    Denneberg, D.: Non-additive measure and integral, The ory and Decision Library Series B: Mathematical and Statistical Methods vol. 27, Klu wer Academic Publish- ers Group, Dordrecht, 1994

    1994

  8. [14]

    L., Rubio de Francia, J

    Garcia Cuerva, J. L., Rubio de Francia, J. L.: Weighted n orm inequalities and related topics, Mathematics Studies 116, North Holland, Amsterdam , 1985

    1985

  9. [15]

    Folland, G.: Real Analysis: modern techniques and appl ications, 2 ndedition, Pure and Applied Mathematics: A Wiley-Interscience Series of Te xt, Monographs, and Tracts, John Wiley& Sons, Inc., New Y ork, 1999

    1999

  10. [16]

    R.: Measure Theory, Graduate Texts in Mathem atics, volume 18, Springer Science+Business Media, New Y ork, 1950

    Halmos, P . R.: Measure Theory, Graduate Texts in Mathem atics, volume 18, Springer Science+Business Media, New Y ork, 1950

    1950

  11. [17]

    Harjulehto, P ., Hurri-Syrj¨ anen, R.: On Choquet integrals and Poincar´ e-Sobolev type inequalities, J. Funct. Anal. 284 (2023), issue 9, paper no. 109862

    2023

  12. [18]

    In: Lenhart, S., Xiao, J

    Harjulehto, P ., Hurri-Syrj¨ anen, R.: Estimates for th e variable order Riesz potential with application. In: Lenhart, S., Xiao, J. (eds.) Potentia ls and Partial Di fferential Equations: The Legacy of David R. Adams, Advances in Analysi s and Geometry vol. 8, De Gruyter, Berl...

    2023

  13. [19]

    Harjulehto, P ., Hurri-Syrj¨ anen, R.: On Choquet integ rals and pointwise estimates, Proceedings of 2022 AWM Research Symposium, Association fo r Women in Math- ematics Series, accepted

    2022

  14. [20]

    Harjulehto, P ., Hurri-Syrj¨ anen, R.: On Choquet integrals and Sobolev type inequal- ities, La Matematica 3 (2024), 1379-1399, doi:10.1007 /s44007-024-00131-z

    2024

  15. [22]

    Harjulehto, P ., H¨ ast¨ o, P ., Latvala, V .: Sobolev embeddings in metric measure spaces with variable dimension, Math. Z. 254 (2006), no. 3, 591–609

    2006

  16. [23]

    H.: On integrals related to and extensi ons of the Lebesgue integrals

    Hildebrandt, T. H.: On integrals related to and extensi ons of the Lebesgue integrals. Bull. Amer . Math. Soc.24 (1917), no. 3, 113–144

    1917

  17. [24]

    arXiv:2311.15224

    Huang, L., Cao, Y ., Y ang, D., Zhuo, C.: Poincar´ e-Sobol ev inequalities based on Choquet-Lorentz integrals with respect to Hausdor ffcontents on bounded John do- mains. arXiv:2311.15224

    arXiv

  18. [25]

    L.: Relative summability, Ann

    Je ffery, R. L.: Relative summability, Ann. of Math. (2) 33 (1932), no. 3, 443–459

    1932

  19. [26]

    Modeling decisions for artificial intelligence, Lecture No tes in Comput

    Kawabe, J.: Convergence in Measure Theorems of the Choq uet Integral Revisited. Modeling decisions for artificial intelligence, Lecture No tes in Comput. Sci. 11676 Lecture Notes in Artificial Intelligence, Springer Cham (20 19) pp. 17–28

  20. [27]

    Kinnunen, J.: Real Analysis, lectures notes, https://sites.google.com/view/jkkinnun/

  21. [28]

    D., Spector, D.: An improvement to the John-Nirenberg ine quality for functions in critical Sobolev spaces

    Mart´ ınez, ´A. D., Spector, D.: An improvement to the John-Nirenberg ine quality for functions in critical Sobolev spaces. Adv. Nonlinear Anal. 10 (2021), 877–894

    2021

  22. [29]

    H., Phuc, N

    Ooi, K. H., Phuc, N. C.: The Hardy-Littlewood maximal fu nction, Choquet inte- grals, and embeddings of Sobolev type, Math. Ann. 382 (2022), 1865–1879

    2022

  23. [30]

    Orobitg, J., V erdera, J.: Choquet integrals, Hausdor ffcontent and the Hardy- Littlewood maximal operator. Bull. London Math. Soc. 30 (1998), 145–150. ON LEBESGUE POINTS AND MEASURABILITY WITH CHOQUET INTEGRAL S 19

    1998

  24. [31]

    C.: Elliptic PDEs, Measures and Capacities, T racts in Mathematics 23, European Mathematical Society, Bad Lagensalza, Germany, 2 016

    Ponce, A. C.: Elliptic PDEs, Measures and Capacities, T racts in Mathematics 23, European Mathematical Society, Bad Lagensalza, Germany, 2 016

  25. [32]

    Ponce, A., Spector, D.: A boxing inequality for the frac tional perimeter. Ann. Sc. Norm. Super . Pisa Cl. Sci.(5) 20 (2020), 107–141

    2020

  26. [33]

    In: Lenhart, S., Xiao, J

    Ponce, A., Spector, D.: Some remarks on Capacitary Inte grals and Measure Theory. In: Lenhart, S., Xiao, J. (eds.) Potentials and Partial Di fferential Equations: The Legacy of David R. Adams, Advances in Analysis and Geometry vol. 8, De Gruyter, Berlin (2023) pp. 127–155

    2023

  27. [34]

    Korean Math

    Saito, H.: Boundedness of the strong maximal operator w ith the Hausdor ffcontent, Bull. Korean Math. Soc. 56 (2019), no. 2, 399–406

    2019

  28. [35]

    Studia Math

    Saito, H., Tanaka, H.: Dual of the Choquet spaces with ge neral Hausdorffcontent. Studia Math. 266 (2022), no. 3, 323–335

    2022

  29. [36]

    Saito, H., Tanaka, H., Watanabe, T.: Abstract dyadic cu bes, maximal operators and Hausdorffcontent. Bull. Sci. Math. 140 (2016), no. 6, 757–773

    2016

  30. [37]

    Positivity 23 (2019), no

    Saito, H., Tanaka, H., Watanabe, T.: Fractional maxima l operators with weighted Hausdorffcontent. Positivity 23 (2019), no. 1, 125–138

    2019

  31. [38]

    Song, L., Xiao, J., Y an, X.: Preduals of quadratic Campa nato spaces associated to operators with heat kernel bounds, Potential. Anal. 41 (2014), no 3, 849–867

    2014

  32. [39]

    Tang, L.: Choquet integrals, weighted Hausdor ffcontent and maximal operators, Georgian Math. J. 18 (2011), 587–596

    2011

  33. [40]

    Gamberini e Parmeggiani, (1905)

    Vitali, G.: Sul Problema della misura dei grupp di punti di una retta, Bologna, Tip. Gamberini e Parmeggiani, (1905)

    1905

  34. [41]

    Watanabe, H.: Estimates of maximal functions by Hausdo rffcontent in a metric space, Advanced Studies in Pure Mathematics 44, 2006, Potential Theory in Matsue, 377-389

    2006

  35. [43]

    Zakon, E.: Integration of non-measurable functions, Canad. Math. Bull. 9 (1966), 307–330

    1966

  36. [44]

    P .: Weakly Di fferentiable Functions, Graduate Texts in Mathematics, 120, Springer-V erlag, New Y ork, 1989

    Ziemer, W . P .: Weakly Di fferentiable Functions, Graduate Texts in Mathematics, 120, Springer-V erlag, New Y ork, 1989. (Petteri Harjulehto) D epartment of Mathematics and Statistics, FI-00014 U niversity of Helsinki, Finland Email address: petteri.harjulehto@helsinki.fi (Rit...

    1989

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.