# Example that does not contradict the Dominated Convergence Theorem

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.

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 $$(0,\infty)$$.