Why Dominated Convergence Theorem is not applicable in this case? Suppose $\omega$ is distributed uniformly over $(0,1]$. Define random variables $$X_n:=n\mathbf{1}_{(0,1/n]}.$$
Obviously, $X_n\rightarrow X=\mathbf{0}$ and $\lim_{n}E[X_n]$ is not equal to E[X]. In this case the Dominated Convergence Theorem (DCT) is not applicable here.
I was wondering how to show that we could not find an integrable random varible Y (i.e. $E[|Y|]<\infty$) such that $|X_n| < |Y| \;a.e.$
At first I think if $Y$ is integrable, then it should be essentially bounded. But afterwards I found it was not true. Then I got stuck... 

Consider $f=\max\limits_{n\in\mathbb{N}}n\mathbf{1}_{(0,1/n]}$, which is given by
$$
f(x)=\begin{array}{}n&\text{if }x\in\left(\frac1{n+1},\frac1n\right]\end{array}
$$
This would be the smallest candidate for a dominating function.
However, notice that
$$
\int_{\frac1{n+1}}^{\frac1n}f(x)\,\mathrm{d}x=\frac1{n+1}
$$
Thus,
$$
\int_0^1f(x)\,\mathrm{d}x
$$
diverges since the Harmonic Series diverges.
