Let $\mu$ be a probability measure on $\mathbb{R}^m$ (so $\int_{\mathbb{R}^m} \mu(d x) = 1$).
Let $f_i:\mathbb{R}^m \rightarrow \mathbb{R}_{\geq 0}$ be integrable functions and let also $\limsup_{i \rightarrow \infty} f_i$ be an integrable function.
Assume that
$ \limsup_{i \rightarrow \infty} \int_{K} f_i(x) \ \mu(d x) \leq \ c \quad \text{ for all compact sets } K \subset \mathbb{R}^m.$
Find under which additional assumptions (for instance on the $f_i$s and/or on $\int f_i$s) we also have:
$ \limsup_{i \rightarrow \infty} \int_{\mathbb{R}^m} f_i(x) \ \mu(d x) \ \leq \ c$
Notes: The measure in the two integrals is the same. What changes is the domain: from $K$ (compact, arbitrarily large) to $\mathbb{R}^m$. $c \in \mathbb{R}_{\geq 0}$.