In this study, we will introduce an innovative and comprehensive multifractal framework, substantiating counterparts to the classical findings in multifractal analysis and We embark on an exploration of the mutual singularity existing between the broader multifractal Hausdorff and packing measures within an expansive framework. An exemplar of this framework involves the application of the ”” multifractal formalism to our core results, elucidating a realm of compact subsets within a self-similar fractal structure — a familiar illustration of an infinite-dimensional metric space. Within this context, we provide estimations for both the overall Hausdorff and packing dimensions. It is noteworthy that these outcomes offer novel validations for theorems underpinning the multifractal formalism, rooted in these comprehensive multifractal measures. Furthermore, these findings remain valid even at points where the multifractal functions governing Hausdorff and packing dimensions diverge. Additionally, we introduce the concepts of lower and upper relative multifractal box dimensions, accompanied by the general Rényi dimensions. A comparison is drawn between these dimensions and the general multifractal Hausdorff dimension, along with the general multifractal pre-packing dimension. Finally, we establish density conclusions pertaining to the multifractal extension of the centered Hausdorff and packing measures. Specifically, we unveil a decomposition theorem akin to Besicovitch’s theorem for these measures. This theorem facilitates a division into two components – one characterized as regular and the other as irregular – thus enabling a targeted analysis of each segment. Following this dissection, we seamlessly reintegrate these segments while preserving their intrinsic density attributes. The impetus behind investigating these comprehensive measures arises when a set holds either a conventional Hausdorff measure of zero or infinity; In such cases, we can identify a function that bestows a positive and finite general Hausdorff measure upon the set , and studies the dimensions of infinitely dimensional sets.

中文翻译：

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新的一般分形测度的多重分形形式主义

在这项研究中，我们将引入一个创新且全面的多重分形框架，证实多重分形分析中经典发现的对应物，并且我们开始探索更广泛的多重分形豪斯多夫和扩展框架内的包装测度之间存在的相互奇点。该框架的一个范例涉及将“多重分形形式主义”应用于我们的核心结果，阐明自相似分形结构内的致密子集领域——这是无限维度量空间的常见图示。在此背景下，我们提供了总体豪斯多夫和包装尺寸的估计。值得注意的是，这些结果为植根于这些综合多重分形度量的多重分形形式主义的定理提供了新颖的验证。此外，即使在控制豪斯多夫和堆积维数的多重分形函数出现分歧的点上，这些发现仍然有效。此外，我们还介绍了下相对多重分形盒维数和上相对多重分形盒维数的概念，以及一般的 Rényi 维数。将这些维度与一般多重分形豪斯多夫维度以及一般多重分形预填充维度进行比较。最后，我们建立了与中心 Hausdorff 的多重分形扩展和堆积测度相关的密度结论。具体来说，我们针对这些措施推出了类似于贝西科维奇定理的分解定理。该定理有助于将其分为两个部分——一个部分具有规则性，另一个部分具有不规则性——从而能够对每个部分进行有针对性的分析。在此解剖之后，我们无缝地重新整合这些片段，同时保留其固有的密度属性。当一个集合拥有零或无穷大的传统豪斯多夫测度时，研究这些综合测度背后的动力就会出现。在这种情况下，我们可以识别一个函数，该函数对集合 赋予正且有限的一般豪斯多夫测度，并研究无限维集合的维数。