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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? enter image description here

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    When $E=[0,1)$, $\chi=1$ on $[0,1)$ and zero everywhere else... is that part of what you were asking? – 2017-02-22
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    Actually in order to understand both examples of how they are going pointwise to 0 , i took the second example as it is easy i think , so that if second is understood , i can apply it to understand first,.... i am looking at 2nd example... – 2017-02-22
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    Gotcha. Answer coming. – 2017-02-22

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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?

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    I am unable to visualize how $g_{n} \rightarrow 0$ pointwise ?? any help!! – 2017-02-22
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    $g_{n}$ gives either 0 or 1 , but how it converges pointwise to 0? – 2017-02-22
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    What is the definition of pointwise convergence you have? – 2017-02-22
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    First we fix f and then through a sequence of functions $f_{n}$ show that $f_{n} \rightarrow f$ for each x .. is this correct? – 2017-02-22
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    $f_{n}:X \rightarrow R$ ,is said to converge point wise to $f$ , if $\{f_ {n}(x )\}$ converges to $f(x)$ for each value of $x \in X$...now?? – 2017-02-22
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    Yes. Pick any $x\in\mathbb{R}$. For large enough $n$, $f(x)=0$ and remains zero for all $n$. That is pointwise convergence to zero. – 2017-02-22
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    as $n \rightarrow \infty , g_{n} = \chi_{(\infty , \infty+1)}$ , then ??.. i know we cant write like that but ... – 2017-02-22
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    i think if you pause and look at it for a moment, you'll see what i mean. you are right that we can't write it like that, because there is no such interval. – 2017-02-22