By introducing a novel expression for the Hausdorff and packing measures with respect to gauge functions, we establish a connection between different approaches to defining the exact Hausdorff and packing dimensions of measures. Furthermore, we define the upper and lower Hausdorff and packing dimensions of a Borel probability measure and demonstrate that these dimensions can be represented using an entropy formula. This paper also addresses an intriguing question raised multiple times regarding the structural representation of singular probability measures. We provide an affirmative response to this question by utilizing characteristic measures.

中文翻译：

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新的测度分形维数和奇异连续测度的分解

通过引入关于规范函数的豪斯多夫和包装测度的新颖表达式，我们在定义精确的豪斯多夫和包装测度维度的不同方法之间建立了联系。此外，我们定义了 Borel 概率测度的上、下 Hausdorff 和堆积维度，并证明这些维度可以使用熵公式表示。本文还解决了多次提出的关于奇异概率度量的结构表示的有趣问题。我们利用特色措施对这个问题做出了肯定的回答。