勒贝格可积函数的强第二积分中值定理及其证明

A New Proof of the Stronger Second Mean Value Theorem for Integrals

中文关键词:  第二积分中值定理  黎曼可积  勒贝格可积

英文关键词:second mean value theorem for integrals  Riemann integrable  Lebesgue integrable

中文摘要:

      R. Witula等人在加额外限制条件下,得到了黎曼积分的强第二积分中值定理.本文在无额外限制条件下得到了相同的结论.同时利用连续函数在$L^p[a,b]~(p \geq 1)$空间的稠密性,将强第二积分中值定理推广到$L^p[a,b]$空间.

英文摘要:

      R. Witula et al obtained a stronger version of the second mean value theorem for integral with some restrictions. In this paper, the stronger version theorem is proved without any restriction. The result is first restricted to the Riemann integrable functions and can be easily generalized to $L^p$ integrable functions by using the well-known result that continuous functions are dense in the Banach space $L^p[a,b]$ for any $p \geq 1$.