Prove $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

Question

Let $\mathcal{Q_k}=[-k,k]^n\subset \mathbb{R^n}$ for all $k\in\mathbb{N}$, the n-dimensional cubes, and $f$ any integrable (lebesgue) function.

Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

I proved it if $f$ is a characteristic or simple function but I can't end the case $f=\lim \phi_k$, where $\phi_k$ are simple. Any idea? Thanks!

Answer 1

Note that, by the monotone convergence theorem,

$$\int_{\cal{Q}_k}|f| = \int \chi_{\cal{Q}_k}|f| \rightarrow \int |f| \quad\, (k\rightarrow\infty)$$

while, on the other hand $$ \int |f| = \int_{\cal{Q}_k}|f| + \int_{\cal{Q}_k^c}|f|$$ for every $k$. So the claim follows from elementary limit considerations.