I have a function
$U(t)=\int_\mathbb{R} u(x) \cos(xt)dx$
and I am trying to use Lebesgue's dominated convergence theorem to show $U(t)$ is continuous for all $t \in \mathbb{R}$
This is the proof.
Notes that $x \to \cos(xt)u(x)$ is measurable for every $t \in \mathbb{R}$
How do we know $x \to \cos(xt)u(x)$? And how do we know it is measurable?
Also note that $|\cos(xt)u(x)| \leq |u(x)|$ and that $|u|$ is by assumptions integrable
I assume this is from the Lebesgue dominated convergence theorem?
Thus $U(t)$ is well defined for every $t \in \mathbb{R}$ Now choose a sequence $t_n \to t$.
Since $\cos(t)$ is continuous we get $\cos(xt_n)u(x) \to \cos(xt)U(x)$ and as $|\cos(xt)u(x)| \leq |u(x)|$
Is this also from Lebesgue dominated convergence theorem?
Hence $U(t_n) \to U(t)$ proving continuity at every $t \in \mathbb{R}$