| Recently,a number of scholars have attracted the attention of the classical Borel-Cantelli Lemma and have made a deep study on Borel-Cantelli Lemma. Especially,by weakening the independent conditions in the second part of Borel-Cantelli Lemma,a lot of results have been perfectly obtained.For example, Erdods and Renyi(1959)have proved that the mutual independent condition on {An,n≥1}can be replaced by the weaker condition of pairwise independence.Renyi (1965)also have given a more general proposition:if (?)P(An)=∞and (?) then P(?). Lamperli(1963)have proved the following propositio successfully:if (?)P(An)=∞and (?) then P(?)>0Kochen,Stone(1964)and Spitzer(1964)successfully have combined the two results above and have extended them into the following proposition: if (?)P(An)=∞,then (?)A lot of scholars have obtained the generalization of second part of the Borel-Cantelli Lemma under weak independent conditions, see for example, Mori and Szekely (1983), Ortega and Wschebor (1983), Petrov (2002,2004), Yan (2006), Amghibech (2006) and Chandra (2008), etc. Xie (2008) obtained a bilateral inequality on the Borel-Cantelli Lemma. But Hu et al. (2009) pointed out there were several mistakes in the proof of the bilateral inequality and example given in Xie (2008), then they presented a corrected version therein. Xie (2009) extended the result of Xie (2008) to the case of nonnegateive bounded sequence of random variables. Feng, et al. (2009) got a weighted version of the Kochen-Stone-Spitzer Theorem.In the paper, we are interested in the results of the Kochen-Stone-Spitzer Theorem and Feng, et al. (2009).The main purpose of the paper is to extend the above result to the case of a sequence of measurable sets defined on a measure space (S, BS,μ) and present a weighted version of the Kochen-Stone-Spitzer Theorem. First, we will give the introduction and the definition of measurable set, measure space, and some inequalities which will be used in the paper. Second, we will give the Borel-Cantelli Lemma and its proof. Hu et al. (2009) pointed out there were several mistakes in the proof of Theorem of Xie (2008). In addition, we will give out some subject about Borel-Cantelli Lemma. Last, we will give the generalization of the Borel-Cantelli Lemma on the measure space and present a weighted version of the Kochen-Stone-Spitzer Theorem. We will also give some examples for results we abtianed. |
