Let {fn} be a sequence of real-valued measurable functions de\x0cned on [0,1].Show that there exists a sequence of positive real numbers {an} such that anfn->0 a.e.

Let {fn} be a sequence of real-valued measurable functions de\x0cned on [0,1].Show that there exists a sequence of positive real numbers {an} such that anfn->0 a.e.
Let {fn} be a sequence of real-valued measurable functions de\x0cned on [0,1].Show that there exists a sequence of positive real numbers {an} such that anfn->0 a.e.

Let {fn} be a sequence of real-valued measurable functions de\x0cned on [0,1].Show that there exists a sequence of positive real numbers {an} such that anfn->0 a.e.
给定n>0,fn^(-1)[-N,N]对所有N=1,2,.的并 = [0,1]
存在 Nn>n 使得 An = fn^(-1)[-Nn ,Nn] 的测度 > 1-1/(2^n)
取 an = 1/Nn^2,于是 对任何 x 属于 fn^(-1)[-Nn ,Nn],anfn(x)= 1 - 1/(2^n)-1/(2^(n+1))-1/(2^(n+2))-...= 1- 2/2^n
并且 对一切 x属于En,m>=n,amfm(x) 0
令E=En 对n=1,2,...求并.
则 任给x属于E,anfn(x) ---->0
同时 E的测度 > En的测度 > 1- 2/2^n -----> 1.
即 E的测度=1.所以结论成立.

英文的？