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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 $\mathbb{R}^m$. $c \in \mathbb{R}_{\geq 0}$.

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    I suspect it's true.2012-03-02
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    @Adam I don't know the site etiquette, but it strikes me as against it to ask a question, accept an answer, then change the question based on the answer (and unaccept answer), but I'll remove my answer as it no longer seems relevant.2012-03-02
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    I'm sorry about that.. I thought opening a new (closely-related) question would have been even worse.. By the way, I did appreciate your answers.2012-03-02
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    I'll undelete because of comments and note that it is no longer an answer.2012-03-02

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