# 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 '10 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)... – Tom Boardman Aug 10 '10 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 '10 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.) – Rasmus Aug 10 '10 at 17:10

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, $$0=\int f dm \not\leq \lim_{n\to\infty}\inf\int f_n dm = -1.$$
For $E=[0,1]$ set: $f_n(x)=n- n^2x$ for $x\in [0,\frac1n]$ $f_n(x)=0$ else.