For a continuous function $f :\mathbb{R}\rightarrow \mathbb{R}$ satisfying $\int_{\mathbb{R}}|f(x)|dx<\infty$, which of the following is true? Problem: For a continuous function $f :\mathbb{R}\rightarrow \mathbb{R}$ satisfying $$\int_{\mathbb{R}}|f(x)|dx<\infty$$ and for some $\alpha >0$ let $d_f(\alpha)$ be the Lebesgue measure of the set $$\{x\in \mathbb{R} : |f(x)|>\alpha\}$$
Then, for all $\alpha \geq 0$ which of the following is true?

*

*$\alpha d_f(\alpha)\leq \int_{\mathbb{R}}|f(x)|dx$

*$\alpha^2 d_f(\alpha)\leq \int_{\mathbb{R}}|f(x)|dx$

*$d_f(\alpha)\leq \alpha \int_{\mathbb{R}}|f(x)|dx$

*$d_f(\alpha)\leq \alpha^2 \int_{\mathbb{R}}|f(x)|dx$
My solution: As $\alpha<|f(x)|$ on $\{x\in \mathbb{R} : |f(x)|>\alpha\}$ we have
$$\int_{\{x: \alpha<|f(x)|\}}\alpha<\int_{\{x: \alpha<|f(x)|\}}|f(x)|<\int_{\mathbb{R}}|f(x)|dx$$
i.e., $$\alpha.d_f(\alpha)<\int_{\mathbb{R}}|f(x)|dx$$
So, I can see that first option is true.
I belive all the other three options are not necessarily true but i could not come up with an example.
Please help me to see if my argument for first bullet is sufficient/clear and help me to see other in detail.
Thank you.

your proof is correct but none of the others are correct. For the last 2 pick a small alpha. For 2nd one, take a big alpha and take a function which decays slower than than alpha squared. Something like 1/sqrt{x} seems a good candidate
