I am having problems with proving the following:
Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that
$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$
I know that the right-hand side value is always finite for each $t \in \mathbb{R}$ due to Chebyshev's inequality. I was also able to prove that
$\lim_{t \to \infty} \mu(x: |f(x)| \geq t) = 0$. But unfortunately can find anything about how fast this value goes to $0$ as $t \to \infty$.
Would be grateful, if you could give an idea and help with this.