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  1. I have a sequence $f_n$ of measurable, positive functions over some common domain $A$. I don't know whether they are increasing, yet I know that they converge to an integrable function $f$ on $A$ for which $f\ge f_n$ for every $f_n$ in my sequence.

Given: Theorem: If $(f_n)$ is increasing and each $f_n$ is positive on $A$ and $f_n \rightarrow f$ pointwise on $A$, except possibly on a subset of measure 0, then $\lim \int_Af_n=\int_A\lim f_n$.

My question, since I don't know whether the sequence in 1) is increasing, am I allowed to arrange the sets to that they are and then apply the theorem? I realize that this is probably more of a logic question than an analysis question.

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up vote 1 down vote accepted

You are trying to apply Montone Convergence, but what you need to apply is Dominated Convergence.

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I thought about that, but I don't think that every Lebesgue integrable function is bounded. – cap Nov 10 '12 at 23:39
$f$ itself is the bound, it needs not be bounded as a function, but integrable. Also, for functions, we don't have nothing like $\lnot(f\le g)\implies f\ge g$, so we can't order them to satisfy $f_1\le f_2\le f_3\le...$ – Berci Nov 10 '12 at 23:46
Consider for example the functions $f:=x\mapsto x^2$ and $g$ the constant $4$. Which one is the bigger? $f\le g$ would mean $f(x)\le g(x)$ for all $x$. – Berci Nov 10 '12 at 23:57

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