测度空间中近可加函数的稳定性  

ON THE STABILITY OF APPROXIMATELY ADDITIVE FUNCTIONS IN MEASURE SPACES

作　　者：楼红卫[1] 

机构地区：[1]宁波大学应用数学系

出　　处：《宁波大学学报（理工版）》1996年第2期1-8,共8页Journal of Ningbo University：Natural Science and Engineering Edition

摘　　要：考虑了测度空间中近可加函数的稳定性，即用可加函数对近可加函数的逼近问题，文中给出了可用可加函数逼近的函数的Ｌ∞、ＬＰ刻划．Suppose f(x) is a function satisfying f(x + y) - f(x) - f(y) = 0,then f(x)is called an additive function. The, stability of the functional equation f(x + y) - f(x)- f(y)=0 was considered. The main results were:Lemma Suppose R is the Lebesgue measure space, R￣2 = R × R is the multiplicative measure space, f(x) is a measurable function,such that f(x +y) -f(x) -f(y) = 0 a.e. (x,y) ∈ R￣2then there exists a unique g(x) such that g(x + y) = g(x) + g(y)andf(x) = g(x) a.e. x ∈ RTheorem 1 Suppose f:R → R is a Lebesgue measurable funcion,such that ‖f(x +y) -f(x) -f(y)‖∞≤δ, i.e.|f(x+y) -f(x) -f(y)| ≤δthen there exists a unique g(x) such that g(x + y) = g(x) + g(y)and|f(x) -g(x)|≤δTheorem 2 Suppose p ∈ (0,+ ∞), f:R → R is a Lebesgue measurable function,such that holds for any y ∈ R,where dx is the Lebesgue measure, then f(x) must be an additive fraction.Theorem 3 Suppose p ∈ (0,+ ∞), f:R → R is a Lebesgue measurable function,such that where dxdy is the multiplicative measure of Lebesgue measure dx and dy. then there exists a unique additive function g(x) such that f(x) = g(x) for almost every x ∈ R.

关 键 词：测度空间 近可加函数 稳定性 

分 类 号：O174.12[理学—数学]