A certain regular property of the method I construction and packing measure (Q2465532)

Let \(\tau\) be a premeasure on a complete separable metric space, then we can induce a measure \(\tau^*\) as follows: \[ \tau^*(B):=\inf\biggl\{\sum_{i=1}^\infty \tau(U_i): \bigcup_iU_i\supset B \biggr\}, \] the measure \(\tau^*\) is called the method I measure of the premeasure \(\tau\) [see \textit{C. A. Rogers}, Hausdorff measures, London: Cambridge University Press (1970; Zbl 0204.37601)]. In the paper under review, under some conditions upon \(\tau\) (the conditions C0--C4 of the paper), the author establishes some relations between \(\tau\) and \(\tau^*\) which could be regarded as the continuation of the previous work of the first author and \textit{Z.-Y. Wen} [Stud. Math, 165, 125--134 (2004; Zbl 1055.28004)]. The main results of the paper are as follows: Theorem 1. \(\tau^*\) has a regularity as follows: Every \(\tau^*\)-measurable set has measure equivalent to the supremum of premeasures of its compact subsets. Theorem 2. The packing measure has above regularity if and only if the corresponding packing premeasure is locally finite. These results, as well as the techniques of the proofs, are interesting for understanding the general method I measure.