遍历积分的计算与推广

Calculation of ergodic integral and its generalization

遍历理论研究的是群在可测空间上作用的定性理论。这里主要分析遍历积分的计算公式,将[0,1]区间上的遍历积分公式推广到实数区间上,分别用A、B两类积分来表示有理区间和无理区间,并给出积分公式,可以看到[0,1]区间上的遍历积分公式是提出A类遍历积分的特例。还给出遍历积分中变量在定义域内做遍历运动的几何表述。

What Ergodic theory studies is mainly about the qualitative property of group actions on measure spaces. In this paper, the formula of ergodic integral is analyzed. The integral on interval [0, 1]is generalized to real number interval, and rational interval and irrational interval are described with class A and class B respectively. Then integral formula of the two classes is given. It can be seen that the [0, 1] interval integral is a special case of the proposed class A. A description of independent variable＇s ergodic movement in its interval by a geometrical way is given.