【摘要】 设E为n维欧氏空间Rn的可测子集,m E<+∞,f(x)为E上的非负可测函数,并记Ek=E{x|k≤f(x)<k+1}(k=0,1,2,…),利用Lebesgue积分的性质,通过构造反例,指出"级数∑k km Ek收敛"并不是f(x)在E上Lebesgue可积的充分条件.进一步,通过增加条件"f在E上a.e.有限",得到相应的充分必要条件.更多还原

【Abstract】 Let E be a measurable subset of Rnwith m E < + ∞. Let f( x) be a non-negative measurable function on E and Ek= E{ x | k ≤ f( x) < k + 1}( k = 0,1,2…). This paper uses a counterexample to show that the convergence of the series ∑k km Ekdoes not imply the integrability of f( x). Furthermore,this paper obtains a necessary and sufficient condition for its integrability.更多还原