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I'm interested in knowing whether there is a condition for general measure spaces under which we know that we can only achieve the strict inequality of Fatou's lemma. I am working in the situation that $f_n \rightarrow f$, and the limit of the integrals do exist so that Fatou's lemma says $ \int f \leq \lim_{n \rightarrow \infty} \int f_n \;. $ Is there a condition on $f_n$ and $f$ which ensures $ \int f < \lim_{n \rightarrow \infty} \int f_n \;. $

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    For measure spaces with total mass $f$inite, look up "uni$f$orm integrability."2011-08-26

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There is a nice discussion of this point in ANALYSIS by Lieb & Loss (section 1.9 of the second edition): It $f_n$ are non-negative and converge a.e. to $f$, then $ \liminf_n\int f_n = \int f +\liminf_n\int|f-f_n| $ provided $\sup_n\int f_n<\infty$.