Let $\mu$ be a probability measure on $X$.
Consider a family of functions $\phi_k: X \rightarrow \mathbb{R}_{\geq 0}$ such that $\sup_k \phi_k(\cdot)$ is integrable over $X$.
Let $\{X_n\}$ be an infinite sequence of compact sets such that $X_n \subset X$, $X_n \subseteq X_{n+1}$ and $X_n \rightarrow X$.
It seems to me that the following implication is true.
$ \int_{X_n}\sup_k \phi_k(x) \mu(dx) \leq \Phi(x) \ \forall X_n \ \Rightarrow \ \int_X \sup_k \phi_k(x) \mu(dx) \leq \Phi(x)$
Is it only a matter of applying the $\lim_{n \to \infty}$ since $\Phi(x)$ is not depending on $n$?
Is such family $\{\phi_k\}$ Uniformly Integrable?