I got how the strict inequality occurs in Fatous lemma but why the limit of the characteristic function $\chi_{(n,n+1)} \rightarrow 0$ pointwise, for $E =\mathbb{R}$ , what happens here when $E =[0,1)$?
How to think about the limit of the function goes to 0 for each value of x?
In your second example, $g_n=\chi_{(n,n+1)}$, so the integral of $g_n$ always equals $1$, since $g_n$ is defined to be $1$ for an interval of length $1$ and zero elsewhere. However, for any $x\in \mathbb{R}$, we may find an $N_x\in\mathbb{N}$ such that for all $n\geq N$, $g_n(x)=0$, which is why we can say that $g_n\to 0$ pointwise. So the integral of the pointwise limit is $0$, yet the integral of $g_n$ for all $n$ is $1$. (Note, all integrals are over all of $\mathbb{R}$.)
Does this help?