Give an example of strict inequality in Fatou's Lemma

Question

The fatou's Lemma says

Let $ \left\{ f_n, n = 1,2,...\right\} $ be a sequence of non-negative measurable functions. Then $$ \liminf \int f_n \geq \int \liminf f_n$$

Some hint?

I think this Showing that the Fatou's lemma inequality can be strict. would work, but I don't understand why $\liminf f_n=0$ in the example he gives.

https://math.stackexchange.com/questions/545215/why-is-there-inequality-in-fatous-lemma?rq=1 — cqfd, May 14 '19 at 05:20

score 2 · Answer 1 · answered May 14 '19 at 05:20

2

$$f_n(x)=\begin{cases}\boldsymbol 1_{[0,1/2]}(x)&n\ \text{odd}\\ \boldsymbol 1_{[1/2,1]}(x)&n\ \text{even}\end{cases}$$

answered May 14 '19 at 05:20

user657324

1 Answers1