Give an example of sequence of Measurable functions defined on some measurable subset $E$ of $\mathbb{R}$ such that $f_{n} \to f$ pointwise almost everywhere on $E$ but $$\int\limits_{E} f \ dm \not\leq \lim_{n} \inf \int\limits_{E} f_{n} \ dm$$
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$\begingroup$ I know that this is very much related to Fatou's lemma but i am unable to find an example. $\endgroup$– anonymousAug 10, 2010 at 13:28
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8$\begingroup$ How is it related to fatou's lemma? What is fatou's lemma? What have you tried? Why do you care? Including such things in your question makes it a pleasure to read (bolstering the site and your reputation) and to answer (since answering, knowing your answer will be useful, makes it all the more worthwhile)... $\endgroup$– Tom BoardmanAug 10, 2010 at 13:32
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$\begingroup$ @Tom Boardman: Doesn't fatou's lemma deal with this type of integral. The final statement of Fatou's lemma is $$\int\limts_{E} f \ dm \leq \lim_{n} \inf \int\limits_{E} f_{n} \ dm$$ Also, whats wrong in thinking that way. Its my think. I did think something about characteristic function, but didn't work. $\endgroup$– anonymousAug 10, 2010 at 15:21
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$\begingroup$ @Chandru1: Maybe you could edit your question and exchange \nleq with \not\leq so that the indented becomes more readable. (As a bounty you'll get an upvote from me.) $\endgroup$– RasmusAug 10, 2010 at 17:10
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2 Answers
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Since Fatou’s Lemma holds for non-negative measurable functions, I suppose you are looking for an example involving not necessarily non-negative functions.
Take, for instance, $f_n(x)=-1_{n\leq x\leq n+1}$ (just $-1$ times the indicator function on the interval $[n,n+1]$). Then $f=\lim\limits_{n\to\infty}\inf f_n=0$. Hence, $$ 0=\int f dm \not\leq \lim_{n\to\infty}\inf\int f_n dm = -1. $$
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$\begingroup$ Thanks. I understood your solution nicely. $\endgroup$– anonymousAug 10, 2010 at 15:58
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$\begingroup$ I was answering the OP's question as it could be read, and in a similarly bald fashion to the way in which the question was asked. Since he had not bothered to fix the Tex or explain the background I read his \nleq as \neq. My attempt to make a point following my initial comment went, it seems, unnoticed. $\endgroup$ Aug 10, 2010 at 16:05
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For $E=[0,1]$ set: $f_n(x)=n- n^2x$ for $x\in [0,\frac1n]$ $f_n(x)=0$ else.
answered Aug 10, 2010 at 13:39
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$\begingroup$ Also, use the open interval if you want a defined pointwise limit. $\endgroup$ Aug 10, 2010 at 14:02
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$\begingroup$ I think that this is an example for strict inequality in Fatou's Lemma. It is not a counterexample to Fatou's Lemma (which cannot get along with non-negative measurable functions, because these obey Fatou's Lemma). $\endgroup$– RasmusAug 10, 2010 at 15:53