Leⅵ逐项积分定理的证明 En={x∈El!(x)≥c0(x) 于是从(应用引理2) X f(x)dx2 f(x)xe,(xdx E E f( fn(x)dx elco(x)dx=cL p(r)idx E E cop(X 得到血n/(x)2c!9(x 令c→则有m[(xk≥[(xh n→>odE 再由的积分定义知mJf(x≥f(x 所以 im.fn(xx f(x)a n->o0 JE ELevi逐项积分定理的证明    → E E n n 得到lim f (x)dx c (x)dx c f x dx x dx E E n n   →  → 令 1,则有lim ( ) ( ) f x dx f x dx E E n n   = → 所以lim ( ) ( ) E {x E | f (x) c (x)} n =  n   f x dx f x dx E n n E lim ( ) ( )    再由的积分定义知 → ( ) ( ) ( ) , ( ) ( ) ( )      =  =  n n n n E E E n E n E E n f x dx c x dx c x dx f x dx f x x dx    于是从（应用引理2） f(x) φ(x) cφ(x) fn (x)