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