Example that does not contradict the Dominated Convergence Theorem

Question

Hi I am doing a past exam paper and I am stuck on one of the questions. Let $\mu=\lambda$ be the Lebesgue measure on $\mathbb{R}$; define the sequence of functions $f_{n}(x)=\frac {1}{n} \chi_{[0,n]}(x):\mathbb{R}\longrightarrow\mathbb{R}$ where $\chi_{[0,n]}$ is the characteristic function of $[0,n]$. Explain why this example does not contradict the Dominated Convergence Theorem. I know $\int\lim_{n\to\infty} f_{n}(x)d\lambda=0$ and $\lim_{n\to\infty}\int f_{n}(x)d\lambda=1$. To show this example does not contradict the Dominated Convergence Theorem I would need to show there is no measurable function $g\geq0$ with $\int g d\lambda<\infty$ that dominates $f_{n}(x)$ but I don't know how to show this. Any help would be appreciated.

Answer 1

If $g$ dominates the $f_n$'s, then on the interval $[n,n+1]$, $g(x)\geq f_{n+1}(x)=\frac{1}{n+1}$. Hence $$\int_0^{n+1}g(x)\,d\lambda\geq\sum_{k=1}^{n+1}\frac{1}{k}$$ so $g$ is not integrable on $(0,\infty)$.