R^(n)上Lebesgue微分定理的证明  

Proof of the Lebesgue differentiation theorem

作　　者：刘锋 李晴 陈碧霞 LIU Feng;LI Qing;CHEN Bixia(College of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410114,China)

机构地区：[1]长沙理工大学数学与统计学院,湖南长沙410114

出　　处：《湖南文理学院学报（自然科学版）》2024年第3期1-6,共6页Journal of Hunan University of Arts and Science(Science and Technology)

基　　金：长沙理工大学实践创新项目(SJCX202117)。

摘　　要：通过分析下半连续函数的定义及其4种等价命题,建立对称差集的测度性质和球可数覆盖定理,证明了球极大函数的下半连续性、可测性及其分布函数的估计不等式,并利用球可数覆盖定理、Lebesgue积分的平移变换定理和Tonelli定理给出了R^(n)上Lebesgue微分定理的详细证明,改进和补充了已有文献的相应结果。By analyzing the definition of lower semcontinuous function and its four equivalent propositions,the measure property of symmetric difference set and the sphere countable covering theorem are established.The lower semicontinuity,measurability and estimation inequality of distribution function of spherical maximal function are proved.The Lebesgue differentiation theorem on R^(n) is proved in detail by using the ball countable covering theorem,the translation transformation theorem of Lebesgue integral and Tonelli theorem,which refine and complement some existing literature.

关 键 词：Lebesgue微分定理 下半连续函数 对称差集 极大函数 

分 类 号：O174.12[理学—数学]