Let $\mu$ be a probability measure on $X \subseteq \mathbb{R}^m$.
Prove that $|| a + B x ||^d$ is integrable iff $||x||^d$ is integrable:
$ \int_{X} || a + B x ||^d \mu(dx) < \infty \ \Leftrightarrow \ \int_{X} ||x ||^d \mu(dx) < \infty $
for any $a \in \mathbb{R}^n$, $B \in \mathbb{R}^{n \times m}$, $d \in \mathbb{R}_{> 0}$
Extend the result to any measurable (non constant) function $f(x)$:
$ \int_{X} || a + B f(x) ||^d \mu(dx) < \infty \ \Leftrightarrow \ \int_{X} || f(x) ||^d \mu(dx) < \infty $