Let $\mu$ be a probability measure on $X \subseteq \mathbb{R}$.
Consider a polynomial function $p_d: \mathbb{R} \rightarrow \mathbb{R}$ of degree $d \in \mathbb{Z}^+$.
I would like to know if the following is true.
$ \int_X |p_d(x)| \mu(dx) < \infty \ \Leftrightarrow \ \int_X |x^d| \mu(dx) < \infty $
In the case the result does work, I'm wondering how it could be extended to the multi-dimensional case $X \subseteq \mathbb{R}^m$, $m \in \mathbb{Z}_{> 1}$.