= Solution
Let $H_0=\int_0^\pi|h(u)|\,du<\infty$. On each period, substitute $x=j\pi+u$ and use periodicity:
$$
\int_{j\pi}^{(j+1)\pi}e^{-\beta x}|h(x)|\,dx
=e^{-\beta j\pi}\int_0^\pi e^{-\beta u}|h(u)|\,du
\leq e^{-\beta j\pi}H_0.
$$
Summing over $j\geq0$ gives the convergent geometric bound
$$
\boxed{\int_0^\infty e^{-\beta x}|h(x)|\,dx\leq\frac{H_0}{1-e^{-\beta\pi}}<\infty.}
$$
Measurability is preserved under multiplication by the exponential, so this proves <Lebesgue integrability>.
Back to article page
