A Note on Measurable Sets without Atoms

DOI：

 作者 单位 曾小林1， 吴明智2 1.重庆工商大学 数学与统计学院， 重庆 400067； 2.中国地质大学（武汉） 数学与物理学院，武汉 430074

针对测度在无原子可测集上的取值问题，提出两个新命题进行讨论。利用下定向的随机变量集合的下确界性质，通过对可测集的剖分，结合可测集与测度的性质，凭借反证法证明了测度为正数〖WTBX〗a的无原子可测集中必存在某个可测子集，使其测度落在区间〖JB（［〗a/3，2a/3〖JB）］〗内；在此基础上用数学归纳法证明了无原子可测集中必有一个单调不增的子集列，使得对无论多小的正实数区间，子集列中总存在某个集，其测度落在区间内；证明了测度不小于某正实数〖WTBX〗λ〖WTBZ〗的无原子可测集中必存在某可测子集，使其测度落在区间〖JB（［〗λ/3，2λ/3〖JB）］〗内；进一步给出经典定理的新证明：若0

Focusing on the range problem of a measure on measurable sets without atoms， we propose two new related propositions for detailed discussion. Utilizing the infimum property of dually directed sets consisting of random variables， via partition of measurable sets， combining with properties of measurable sets and measures， we first prove in a measurable set with positive measure a and without atoms that there exists some measurable subset such that its measure lies in the interval 〖JB（［〗a/3，2a/3〖JB）］〗 by way of contradiction. Next based on the previous result we show that in a measurable set without atoms there must exist a non increasing sequence of measurable subsets such that for any real interval of positive numbers no matter how small， there exists some element in the sequence with its measure in the interval. Then， we prove in a measurable set with measure greater than or equal to some positive number λ and without atoms that there exists some measurable subset such that its measure lies in the interval 〖JB（［〗λ/3，2λ/3〖JB）］〗. Finally， we give a new proof for the classical fact： if 0