可测函数的一些刻画

Some Characterizations on Measurable Functions

本文给出了可测函数的一些刻画,证明了定义在可测集ER n上几乎处处有限的函数f(x)在E上可测当且仅当任给δ>0,存在可测集FE,使得m(EF)<δ且f(x)是F上可测函数.这一结果对经典的卢津定理的逆定理给出了一个实质性的改进.

In this paper, some characterizations on measurable functions are given. The author prove that an almost everywhere finite function f defined on a measurable set E is measurable if and only if for arbitrary δ 〉 0, there is a measurable set F 包含 E, such that m（E- F） 〈 δ and f（x） is a measurable function on F, which improve the inverse theorem of ЛУЗИН-theorem.