Real and complex analysis

26 REAL AND COMPLEX ANALYSISThat (3) holds if ex ~ 0 follows from Proposition 1. 24(c). It is easy toverify that (3) holds if ex = -1, using relations like (-u) + = U -. The caseex = i is also easy: Iff = u + iv, thenf (if) = f (iu - v) = f (-v) + i f u = - f v + i f u = {f u + i f v)=if!Combining these cases with (2), we obtain (3) for any complex ex.IIII1.33 Theorem Iff E L 1 (Jl), thenPROOF Put z = Ix f dJl. Since z is a complex number, there is a complexnumber ex, with 1u :::; 1 exf 1 = 1 f I· Henceex 1 = 1, such that exz = 1 z I. Let u be the real part of ex! ThenThe third of the above equalities holds since the preceding ones show thatJ exf dJl is real.IIIIWe conclude this section with another important convergence theorem.1.34 Lebesgue's Dominated Convergence Theorem Suppose Un} is a sequenceof complex measurable functions on X such thatf(x) = lim fn(x) (1)n-+ coexists for every x E X. If there is afunction 9 E I!(Jl) such that1 f,,(x) 1 :::; g(x) (n = 1, 2, 3, ... ; X EX), (2)and(3)lim r f" dJl = r f dJl.n-+co Jx Jx(4)