利用$(0,1)$中的均匀分布序列计算勒贝格积分
Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$
摘要：
利用$(0,1)$中的均匀分布序列，给出了计算勒贝格积分的Kolmogorov强大数定律的修正证明.推广了C. J. Schoi $β$ engeier（cf. \cite{BaxSch2002}，定理1，第271页）到均匀分布（在$(0,1)$中）序列的最大集合$S_f \subset(0,1)^{\infty}$，该集合严格包含具有无理数$α$的形式为$(\{αn\})_{n \in {\bf N}}$的序列集合，并且对于该集合为$\ell_1^{\infty}(S_f)=1$，其中$\ell_1^{\infty}$表示$(0,1)$中线性勒贝格测度$\ell_1$的无穷次幂。
英文摘要：
We present modified proof of a certain version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$β$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in $(0,1)$) sequences $S_f \subset(0,1)^{\infty}$ which strictly contains the set of sequences of the form $(\{αn\})_{n \in {\bf N}}$ with irrational number $α$ and for which $\ell_1^{\infty}(S_f)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue measure $\ell_1$ in $(0,1)$.
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