非负可积函数的积分

On Integrals of Nonnegative Integrable Functions

设f(x)是[a,b]上非负可积函数.可证integral from a to b(f(x)dx=0)当且仅当{x∈[a,b]:f(x)=0}是[a,b]的稠子集,此外还可得出integral from a to b(f(x)dx>0的一个充要条件.

Let f（x） be a nonnegative integrable function on [a,b]. In this paper, we prove that∫a bf（x）dx iff the set {x∈[a,b]：f（x）=0} is a dense subset of [a,b]. As a corollary of this result, a necessary and sufficient condition for ∫a bf（x）dx〉0 is given.