# Extension of Riemann-Lebesgue Lemma

I'm working my way through Billingsley's Probability and Measure, and I've got a question on Problem 26.5: Show by Theorem 26.1 (Riemann-Lebesgue Theorem) and integration by parts that if $\mu$ has a density $f$ with integrable derivative $f'$ then $\phi(t)=o(t^{-1})$ as $|t|\rightarrow\infty$. Here $\phi(t)$ is the characteristic function.

Integration by parts gives me $$\phi(t)=\int_{-\infty}^\infty f(x)e^{itx}dx=\left[f(x)e^{itx}/it\right]^\infty_{-\infty}-\int_{-\infty}^\infty f'(x)e^{itx}/it dx$$

Multiplying both sides by $t$ then gives $t\phi(t)=-\left[f(x)e^{itx}i\right]^\infty_{-\infty}+\int_{-\infty}^\infty f'(x)e^{itx}i dx$$We want to show the right hand side goes to 0 as$t\rightarrow\infty$. The second term goes to 0 as$|t|\rightarrow\infty$since$f'$is integrable (for this we can use the same argument used to prove the Riemann-Lebesgue Lemma). But how about the first term? Do we know that$f(x)\rightarrow 0$as$|x|$goes to$\infty$? I know that generally for$f(x)$to be integrable this is not required. Is it required if$f'$is integrable? If so, can somebody please give a hint as to why this is so? Thanks! ## 1 Answer According to some form of the fundamental theorem of calculus (Theorem 7.21 of Rudin's Real and Complex analysis), if$f$is differentiable and its derivative is integrable, then$f(x) = f(0) + \int_0^x f'(t)dt$, which converges (by dominated convergence theorem for instance) to$f(0)+ \int_0^{\pm\infty} f'(t)dt \in \mathbb R$when$x$goes to$\pm\infty$. These two limits must be$0$otherwise$f$couldn't be a density. • I don't understand why these limits must be 0. See link for example. – gruyb Sep 15 '16 at 9:29 • In your case, the fundamental theorem of calculus implies that there is a limit in$+\infty$(unlike in the counterexample you mentioned). Now if this limit, which we call$\lambda$, is not 0, then there is no way$f$could integrate to$1$. Hence$\lambda = 0$. Same reasoning for$-\infty\$. – justt Sep 15 '16 at 9:34
• You're welcome ! – justt Sep 15 '16 at 9:52