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If $\{f_n\}$ is a sequence of non-negative Lebesgue measurable function on $\mathbb{R}$, $f_n\longrightarrow f$ a.e., then

$\textbf{(1.) Fatous Lemma}: $ $\int \lim\inf f_n\leq \lim\inf \int f_n$.

I want to see the importance of limit and infimum in the inequalities. Therefore, I would like to consider following natural questions:

With the same hypothesis, do we have the following inequalities?

$\textbf{(2.)}$ $\int \lim f_n \leq \lim\int f_n$

$\textbf{(3.)}$ $\int \inf f_n \leq \inf\int f_n$

Is the conclusion (1) stronger than (2) and (3)? If yes, how? (any example?)

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(2.) holds provided the $\lim f_n$ and $\lim \int f_n$ exist, because in this case they are equal to $\liminf f_n$ and $\liminf\int f_n$.

(3.) also holds since $\inf f_n\le f_n$, so $\int \inf f_n\le \int f_n$ for all $n$, implying $\int \inf f_n\le \inf\int f_n$ (when $t\le s$ for all $s\in S$, then $t\le \inf S)$.

(1.) is a stronger statement that (2.), since it says something is true even when the limits in question don't exist. Neither of (1.) or (3.) implies the other, since inf and lim inf are not very related to each other: the former can be affected when finitely many terms of the sequence are changed, while the latter cannot.

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  • $\begingroup$ Thanks for the clarification. $\endgroup$
    – Groups
    Nov 3, 2014 at 3:20

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