弱收敛在勒贝格积分中存在性证明及其具体应用

为了证明勒贝格积分是否具有弱收敛性,基于勒贝格相关理论,得到勒贝格积分存在弱收敛的充要条件为{f_k}在L^p空间中有界;同时,得出需满足{f_k}在测度E范围内的积分极限值等于其积分值的条件.最后,将勒贝格积分应用在概率统计方面,并采用Lebesgue-Stieltjes积分分别表示随机变量及数学期望.

Existence Proof of Weak Convergence in Lebesgue Integral and Its Application

In order to prove the Lebesgue integral is of weak convergence, basing on Lebesgue theory, we prove the necessary and sufficient conditions for the existence of the weak convergence of Lebesgue integral sequence {f_k} are that the sequence are bounded in L^p space; at the same time, it must satisfy that the integral limit value of {f_k} in the measure range of E is equal to the integral value. Finally, by the application of Lebesgue integral to probability statistics, we use Lebesgue-Stieltjes integral to represent the random variables and mathematical expectation respectively.