Consider $z \in \mathbb{R}^n$ and a sequence $\{ z_i \}_{i=1}^{\infty}$ such that $z_i \rightarrow z$.
Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded.
I wonder if the proposition
$$ \limsup_{i \rightarrow \infty} \phi(z_i, x) \text{ bounded almost everywhere on } X $$
is equivalent to
$$ \exists \delta > 0 \text{ such that: } \sup_{\epsilon \in \delta \mathbb{B}} \phi(z + \epsilon, x) \text{ bounded almost everywhere on } X $$
