复准广义权的全变差

Total Variations of Complex Pre-generalized Weights

引人复准广义权及其全变差的概念，并给出了全变差具有可数可加性的条件．讨论了复准广义权所诱导的两函数空间之间的映照，给出与该映照及全变差二者相关的积分表示．

Ler(X,τ)be a topology space. Let(Y,)be a Hausdorff space,υa posi-tive finite Borel measure on Y. Introduce the notions of the complex (pre-)generalized weightw:(Y)and its total variation |W|, where(Y)is a family of complex functions on Y.Let|W|(X)∈L￣1(υ).If for any ε>0,there exists a compact sudset Y_εof Y such that|W|(x)dυ<ε,thenψ_M(A):=|W (A)|dυ is a weight such that |ψ_M|(A)=|W|(A)dυ,which is a measure on;furthermore, we have f d| ψM|=|w|(f)dυ for any non-negative |ψ_M|-measuradle function f on X ,whene |W|(f)is the υ-measuradle function on Y with respect to f and |W|.