Assume $X$ is a measure space and $f : X \to [0,\infty]$ is an absolutely integrable function (that is $\int_X f \; d \mu < \infty$). This question is about the asymptotic behaviour of the function $$E(\delta) = \int _{f \leq \delta} f \; d \mu.$$

Since $f$ is absolutely integrable, it follows from the downward monotone convergence theorem that $E(\delta) \to 0$ as $\delta \to 0$. What I want to understand is how fast $E(\delta)$ converges to 0.

What I know is that if $\mu(X) < \infty$ then we have $E(\delta) \leq \delta \mu(X)$, so $E(\delta)$ converges to $0$ at a linear rate or faster.

Question: What can be said when $\mu(X) = \infty$?