Let $f \in L^1[0,1]$. Does it hold that $\lim_{t\to\infty}t\cdot|\{x:|f(x)|>t\}|=0$?

Question

Let $f \in L^1[0,1]$.

Does it hold that $$\lim_{t\to\infty}t\cdot|\{x:|f(x)|>t\}|=0\,?$$ I don't know if this is true or not.

Answer 1

Yes, its true, just note that for $t>0$

$$ 0\leqslant t \cdot \lambda (\{|f|>t\})\leqslant \int_{\{|f|>t\}}|f| \,d \lambda \tag1 $$

And because $f$ is integrable

$$ \int_{\{|f|>t\}}|f| \,d \lambda=\int_{\operatorname{dom}(f)}|f|\,d \lambda-\int_{\operatorname{dom}(f)}|f|\mathbf{1}_{\{|f|\leqslant t\}}\,d \lambda \tag2 $$ Now, as $|f|\mathbf{1}_{\{|f|\leqslant t\}}\uparrow |f|$ pointwise as $t \to \infty $ we can apply the monotone convergence theorem to (2), showing that $\int_{\{|f|>t\}}|f|\,d \lambda \to 0$ as $t \to \infty $. Then taking the limit as $t \to \infty $ in (1) we had shown the result.∎