Which counterexamples to keep in mind when studying (Lebesgue) integral convergence theorems When studying the classical theorems regarding the convergence of a sequence of integrals, to the integral of the limit function, e.g.


*

*Beppo Levi

*Dominated convergence 

*Vitalli's theorem
which counterexamples should one keep in mind? 
 A: Here are my favorite counterexamples:


*

*On $\mathbb{R}$, $f_n = 1_{[n,n+1]}$.  This sequence converges to 0 pointwise but the integrals converge  to 1.

*On $\mathbb{R}$, $f_n = \frac{1}{n} 1_{[0,n]}$.  This sequence converges to $0$ uniformly but the integrals converge to $1.$

*On $[0,1]$, $f_n = n 1_{(0, 1/n)}$.  This sequence converges to $0$ pointwise but the integrals converge to $1.$
Any of these examples can be modified to give a sequence where the integrals converge to infinity instead.  You can turn them into a Beppo-Levi counterexample by considering $g_n = f_n - f_{n-1}$, so that $\sum g_n = \lim f_n$ as a telescoping sum.
Also useful to keep in mind:


*

*On $\mathbb{R}$, $f_n = 1_{[n, n+1/n]}$.  This sequence converges to 0 pointwise and in $L^1$ but has no dominating function.

*On $\mathbb{N}$, the sequence $f_n = 1_{\{n\}}$.  This sequence is $L^1$ bounded, convergent pointwise, and uniformly absolutely continuous (try $\delta = 1/2$), but not uniformly integrable (depending on your exact definitions).

*The typewriter sequence which converges to 0 in $L^1$ but diverges pointwise everywhere.

*The sequence $f_n(x) = e^{2 \pi i n x}$ on $[0,1]$, which diverges pointwise and in $L^p$ but converges weakly in $L^2$ to $0$.
A: $$
f_n(x) = \begin{cases} n & \text{if } 0<x<1/n, \\ 0 & \text{if } x\le 0 \text{ or } x\ge 1/n. \end{cases}
$$
Then you have
$$
\lim_{n\to\infty} \int_{\mathbb R} f_n(x)\,dx = 1 \ne 0 = \int_{\mathbb R} 
\lim_{n\to\infty} f_n(x) \,dx.
$$
The reason this doesn't contradict the dominated convergence theorem is of course that
$$
\int_{\mathbb R} \sup_n f_n(x) \,dx = +\infty,
$$
so there is no dominating function whose integral is finite.
