Suppose the below are true :
(1) Functions:
$f_n \to f$ a.e , whic means $f_n$ converges almost everywhere to function $f$ , also
$g_n \to g$ a.e
$h_n \to h$ a.e
2) $g_n$ , $h_n$ and $f_n$ are integrable for all $n \ge 1$
$g$ , $h$ and $f$ are integrable , also
$\int g_n dμ \to \int g dμ$ , $\int f_n dμ \to \int f dμ$ , $\int h_n dμ \to \int h dμ$
3) $g_n \le f_n \le h_n$ a.e
Given the above 1, 2, 3 Hypotheses are true .
How can we find an example where the Hypotheses Dominated Convergence Theorem Fails to hold
Could you please also prove to me and explain why the specific example works?
Take $f_n=g_n=h_n=nE_{[2^n,2^n+\frac{1}{n^2}]}(x)$, where $E_{\circ}$ is the characteristic function, then you see that $f_n,g_n,h_n$ converge to $0$ a.e., all your hypotheses (1), (2), (3) are satisfied, but there exists no dominating integrable bound $h$ for all $f_n$, since otherwise we must have $$h\geq \sum_{n=1}^\infty n E_{[2^n,2^n+\frac{1}{n^2}]}$$ and the integral of the r.h.s. is $\sum_{n=1}^\infty\frac{1}{n}=\infty$. Thus the Lebesgue DCT is not applicable.