I have been studying for a preliminary exam using old exams. Many of them ask the following question which we have unfortunately not talked about in the class at all. It would be nice to see an example of how the Riemann-Lebesgue lemma is applied.
$\textbf{Question:}$ Show that $f(t) = \int_E \sin(tx) dx$ is a continuous function where $E$ is a set of finite measure.
From what I read on wikipedia, the Riemann-Lebesgue theorem deals with Fourier transform of a function, I am not sure how to related this to the above problem.
Let $g(x) = \chi_E(x)$. Note that $g\in L^1(\Bbb R)$ since it is just a characteristic function for a set with finite measure. Then we have that the Fourier transform of $g$ is (up to scalar multiple and conjugation depending on your definition of the Fourier transform)
$$\mathcal{F}g(x) = \int_{\Bbb R} e^{itx}g(t)\,dt.$$
Taking the imaginary part, we get $\int_E \sin(xt)g(t)\,dt.$ Since $g\in L^1(\Bbb R)$, the Riemann-Lebesgue lemma says that the Fourier transform of $g$ is continuous (and moreover it decays at infinity). A complex-valued function is continuous if and only if its component pieces are separately continuous. This supplies the proof.