Show that the following set is a null set

Question

Consider $(f_n)_{n \ge 0} $ sequence of integrable functions on $\mathbb{R}$.

Let $ \lim_{n \rightarrow \infty } \int_{-\infty}^{\infty} |f_n(x)| dx =0$. Let $\alpha > 0$. We define $D_n $ := { $ x \in \mathbb{R} | |f_n(x)| > \alpha $}.

Show that: $ \lim_{n \rightarrow \infty }\lambda(D_n) = 0$. ($\lambda$ is the lebesgue-measure).

The question was from a past exam paper that I am using to study for my upcoming exam. I hope you can help me. Thank you for your reply.

Answer 1

$|f_n| > \alpha \chi_{D_n} $

both are positive so : $ 0 \leq \int_{\mathbb{R}} \alpha \chi_{D_n} d \lambda < \int_{\mathbb{R}} |f_n| d \lambda$

by taking $n$ to $ + \infty$ we have that $ \int_{\mathbb{R}} \alpha \chi_{D_n} d \lambda \to 0$ which is equivalent to $ \alpha \lambda(D_n) \to 0$

and since $\alpha > 0$ then $ \lambda(D_n) \to 0$

Answer 2

One has $$0\leq \int_{D_n} \alpha \, d\lambda<\int_{D_n} |f_n|\, d\lambda\leq \int_\mathbb{R} |f_n|\, d\lambda\to 0 $$ And by the squeeze theorem one has..?