Let $X \subseteq \mathbb{R}^n$ be an unbounded set and let $p(x)$ be a probability density function on $X$, so that $\int_X p(x) dx = 1$.
Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $\phi: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ such that $\phi(\cdot)$ is continuous, $\phi(0) = 0$ and $\lim_{|y| \rightarrow \infty} \phi(y) = \infty$.
$f(\cdot)$ and $\phi(\cdot)$ are such that
$ \int_X \phi(f(x)) p(x) dx < \infty $
I'm wondering if there exists $\epsilon \in \mathbb{R}^n \setminus \{0\}$ such that
$ \int_X \phi(f(x) + \epsilon) p(x) dx < \infty $