real-analysis-class22 实变函数论应用数学讲义.pdf

實變函數論─應用數學系 吳培元老師 Class 22 Sec. 2.10 Applications of DCT. (easier to apply than def. of f ). Thm.f meas.,g integrable. f g a.e. f integrable. Note1: As comparison test for series or improper integral 2: If f simple, then Ex.2.7.2. 3: False for Riemann integrals: 1 if x rational  Ex.f (x )  on [0,1] 0 if x irrational  Then f  1 on [0,1] But f not Riemann integrable Pf: Check: f integrable simple f n  0 f n  f a.e. f n f g a.e. f n simple Ex. 2.7.2 f n integrable DCT f integrable Def.f meas. func. f is essentially bdd if c 0  f c a.e. Def. ess. supf inf c : f c a.e.   Cor 1.f integrable,g meas., essentially bdd fg integrable Note: g may not be integrable Pf: Say, g c a.e. fg c f a.e.  integrable  fg integrable fg integrable Cor 2. E a , u ( E)  f meas., esentially bdd on E  f exists. E Pf: f c a.e. on E .    f c a.e. E E  integrable  f integrable, i.e.,  f exists. E E 1 實變函數論─應用數學系 吳培元老師 (Note:f bdd on finite measure set f integrable Much more general tha