I am asked the following question:
By suitable example, show that strict inequality holds in Fatou's lemma
I know that Fatou's lemma states that
Suppose $f_n$ is a sequence of measurable functions with $f_n$ non-negative. If $f_n$ coverges to $f$ for a.e $x$ then $\int f=\liminf\int f_n$.
I need to construct an example to show that strict inequality holds in lemma.
Please help. I don't have idea to start.
Best Answer
Actually, Fatou's lemma asserts that if $\{f_n\}$ is non-negative, then $$\int_X\liminf_{n\to\infty}f_n\leq \liminf_{n\to\infty}\int_Xf_n $$
To show that strict inequality can hold, consider $f_n=1_{[n,n+1]}$ on $\mathbb{R}$ with Lebesgue measure.