Mcshane积分的LSRS收敛定理

LSRS Convergence Theorem of Mcshane Integral

利用函数的局部小黎曼和性质(LSRS)和函数列的一致局部小黎曼和性质(ULSRS), 建立了Mcshane积分的局部小黎曼和收敛定理(定理1)．然后,通过定理2证明了Lebesgue积分的控制收敛定理为该定理的一个推论．

This paper gains the convergence theorem （Theorem 1） of locally small Riemann sum of Mcshane integral, which utilizes the properties of locally small Riemann sum of function（LSRS） and uniformly locally small Riemann sum of sequence of functions（ULSRS）. Then it is proved that, Lebesgue integral dominated convergence theorem is a corollary of Theorem 1 in Theorem 2.