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Let $(X,M,\mu)$ be a measure space. For any $\{E_n\}_{n=1}^{\infty} \subset M$ which conditions must $\mu$ have in order to satisfy

\begin{equation} \lim\sup\mu(E_n)\leq\mu(\cap_n\cup_{m\geq n}E_m)? \end{equation}

For example if $\mu$ is finite, then it is clear by the continuity from above.

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You need some form of finiteness (of the measure) or boundness on the sequence $\{ E_n \}$.

This is just a form of generalized Fatou's lemma: If there exists an integrable majorant $g$ such that the sequence (of functions) $\{f_n\}$ are bounded by $g$ i.e. $|f_n(x)|\leq g(x)$ then we have: $$\int \liminf f_n \leq \liminf \int f_n \leq \limsup \int f_n \leq \int \limsup f_n $$

Write $f_n = 1_{E_n}$ (the indictaor function) and then you have: $$\limsup \mu (E_n ) \leq \mu\big(\cap_{n\geq 1} \cup_{m\geq n} E_m\big)$$

The last inequality (with the $\limsup$) won't work if there isn't an integrable majorant $g$.

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