Limit of a certain Lebesgue integral

Question

Can someone help me to show that

$\lim_{t \to \infty} \int_{\Bbb R} f(x)\sin(xt)dx =0$ for any Lebesgue integrable function $f$?

Side note: How do you make the $t \to \infty$ appear directly under the "lim" in Mathjax?

Answer 1

This is a question which comes up frequently in measure theory courses, and a style of answer which comes up frequently as well. The idea is this: let $\Gamma$ be the set of $L^1$ functions for which the statement holds. You want to show $\Gamma = L^1$. The best way to do this is to show that $\Gamma$ is closed (with respect to the $L^1$ norm), and then to show that $\Gamma$ contains some dense subset of $L^1$. It's useful to have an array of dense subsets of $L^1$ in the back of your head: simple functions, bounded functions, step functions, continuous functions, smooth functions with compact support, etc. The one you'll want to use here is step functions (linear combinations of characteristic functions of bounded intervals). If you didn't know that step functions were dense in $L^1$, try to prove that as well (approximate simple functions with step functions; simple functions are dense in $L^1$ by definition). Proving the claim for step functions is then very easy.

Answer 2

The fact that the set $\Gamma$ defined in the above answer is indeed closed, follows from the dominated convergence theorem, by noting that the absolute value of the integrand is bounded by $L^1$ function $|f|$.