I know that Fatou's lemma states that if $\{f_j\}_{j\geq 1}$ is a sequence of nonnegative measurable functions and $f_j(x)\longrightarrow f(x)$ almost everywhere on a set E, then \begin{eqnarray} \int\limits_E f\leq \liminf\int\limits_E f_j\notag \end{eqnarray}
Now I have this question: Consider $\mathbb{R}$ equipped with Lebesgue measure. For $j=1,2,\cdots,$ let \begin{eqnarray} f_j(x)=\begin{cases}1+\sin jx & \mathrm{if}~~ -\pi \leq x\leq \pi \\0 & \mathrm{if}~ ~ x<- \pi \\ 0 & \mathrm{if}~~ x> \pi. \end{cases}\notag \end{eqnarray} What does Fatou's lemma say about these $f_j$?
Can somebody assist me here please. I don't know how to begin.
(This is essentially the same as the first comment on the question.)
Let $f = \liminf_{j} f_{j}$. Then by Fatou's lemma $$ \int f \leq \lim_{j} \int f_{j}. $$ You can easily calculate $\int f_{j}$ as well as the right-hand side above. For the left-hand side, you need to figure out what $f$ is. But without doing that, you know an upper bound on $\int f$ from the inequality above, and you can show certain properties of $f$. For example, $f$ cannot be equal to $2$ a.e. on $(-\pi,\pi)$.