Course in Probability Theory

198 1 CHARACTERISTIC FUNCTIONMore generally, we can define the Laplace transform of an s .d .f. or a functionG of bounded variation satisfying certain "growth condition" at infinity . Inparticular, if F is an s .d .f., the Laplace transform of its indefinite integralis finite for ~ . > 0 and given byXG(x) = / F(u) du0G(~ ) _ e--xF(x) dx = fe-~ dxf/.c(d y)4[O,oo) [O,oo) [O,x]_fa(dy) fe-Xx dx = - f e_ ~,cc(dy)[O, ) ) [O oo) Awhereµ is the s .p.m . of F . The calculation above, based on Fubini's theorem,replaces a familiar "integration by parts" . However, the reader should bewareof the latter operation . For instance, according to the usual definition, asgiven in Rudin [1] (and several other standard textbooks!), the value of theRiemann-Stieltjes integralis 0 rather than 1, but1000e-~'x d6o(x)or e-~ d8o (x) = Jim 80 (x)e-; x IIE + f00AToc80 (x)Ae - ~ dxis correct only if the left member is taken in the Lebesgue-Stieltjes sense, asis always done in this book .There are obvious analogues of propositions (i) to (v) of Sec . 6 .1 .However, the inversion formula requiring complex integration will be omittedand the uniqueness theorem, due to Lerch, will be proved by a different method(cf. Exercise 12 of Sec . 6 .2) .Theorem 6 .6 .2 . Let F j be the Laplace transform of the d .f . F j with supportin °+,j=1,2 .If F1 = F2, then F1=F2 .PROOF . We shall apply the Stone-Weierstrass theorem to the algebragenerated by the family of functions {e - ~` X, A > 0}, defined on the closedpositive real line : 7+ = [0, oo], namely the one point compactification ofM,+ = [0, oo) . A continuous function of x on R + is one that is continuousin ~+ and has a finite limit as x --* oo . This family separates points on A' +and vanishes at no point of ~f? + (at the point +oo, the member e-0x = 1 of thefamily does not vanish!) . Hence the set of polynomials in the members of the