# Fatou's Lemma Counterexample

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 $$\int\limits_{E} f \ dm \not\leq \lim_{n} \inf \int\limits_{E} f_{n} \ dm$$

• I know that this is very much related to Fatou's lemma but i am unable to find an example.
– anonymous
Aug 10, 2010 at 13:28
• 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)... Aug 10, 2010 at 13:32
• @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.
– anonymous
Aug 10, 2010 at 15:21
• @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.) Aug 10, 2010 at 17:10

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, $$0=\int f dm \not\leq \lim_{n\to\infty}\inf\int f_n dm = -1.$$

• Thanks. I understood your solution nicely.
– anonymous
Aug 10, 2010 at 15:58
• 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. Aug 10, 2010 at 16:05

For $E=[0,1]$ set: $f_n(x)=n- n^2x$ for $x\in [0,\frac1n]$ $f_n(x)=0$ else.

• n*exp(-nx) will also do the job. Aug 10, 2010 at 13:54
• Also, use the open interval if you want a defined pointwise limit. Aug 10, 2010 at 14:02
• 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). Aug 10, 2010 at 15:53