关于函数方程的可测解

On Measurable Solution of Functional Equations

可测函数类比连续函数类包含更多的函数.已知的结果是在连续函数类内,线性函数、指数函数、余弦函数、正弦函数可以分别被一组函数方程特征刻画.利用勒贝格积分的绝对连续性、勒贝格微分定理及卢津定理,证明在可测函数类内这些特征刻画仍然成立.

The class of measurable functions contains more functions than the class of continuous functions.The known result is that within the class of continuous functions,basic elementary functions such as linear function,exponential function,cosine function,and sine function can be characterized by a set of functional e-quations respectively.Using the absolute continuity of Lebesgue integrals,Lebesgue’s differential theorem and Luzin’s theorem,it can be proved that these characterizations still hold true within the class of measurable functions.