Give an example of a sequence $\{f_n\}_{n=1}^\infty$ of integrable functions on $\mathbb{R}$ such that $f_n \to f$ but $\int f_n \not\to \int f$. Explain why your example does not conflict with the Dominated Convergence Theorem.
I do notice that the inequality $|f_n(x)| \le g(x)$, where $g$ is an integrable function over $\mathbb{R}$, is not listed here in this problem. So the function need not be dominated by an integrable function. But this is required as one hypothesis of the Dominated Convergence Theorem; hence the example will not conflict.
If this is sound reasoning, how may I come up with functions that are not dominated by another function? Initially I was thinking $f_n(x)=x \sin (nx)$ because its $\lim \sup$ is $\infty$, but even then, we still have $|f_n(x)| \le |x| =: g(x)$.
Best Answer
Consider the sequence of functions on $(0,1)$ $$f_n(x) = \begin{cases} n & \text{ if } x \in (0,1/n)\\ 0 & \text{ otherwise} \end{cases}$$ We have $\lim_{n \to \infty} f_n(x) = 0 = f(x)$. However, $$\lim_{n \to \infty} \int_0^1f_n(x)dx = 1 \neq 0 = \int_0^1 f(x) dx$$ The key in the dominated convergence theorem is that sequence of functions $f_n(x)$ must be dominated by a function $g(x)$, which is also integrable.