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. 



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A question about the tail of an absolutely integrable function
You can't say anything in general. For a simple example showing that you can't get any polynomial rate, consider the measure space $X = [1, \infty)$ with the Lebesgue measure and the function $f(x) = x^{-(1 + \epsilon)}$. Assuming I integrated it correctly, you get $E(\delta) = \frac{1}{\epsilon} \delta^{\epsilon/(1 + \epsilon)}$ which gets arbitrarily slow as $\epsilon \to 0$.


You can also do a more complicated construction that gives a bit more: for any function $e(\delta)$ with $e(\delta) \to 0$, you can find an $f$ such that $E(\delta\_k) \ge e(\delta\_k)$
for some sequence $\delta\_k \to 0$. In particular, you can get arbitrarily slow rates. For this construction, choose a decreasing sequence $\delta\_k \to 0$ with $e(\delta\_k) \le 2^{-k}$. Then define $f$ such that $\mu(f = \delta\_k) = 2^{-k}/\delta\_k$. (If $X$ is $[0, \infty)$ with Lebesgue measure, you can do this by setting
$f(x) = \delta\_k$ if $\sum\_{j \le k-1} 2^{-j}/\delta\_j \le x < \sum\_{j \le k} 2^{-j}/\delta\_j$.) Then
$f$ is integrable because
$$
\int f = \sum\_{k} \delta\_k \mu(f = \delta\_k) = \sum\_k 2^{-k} < \infty.
$$
But you have
$$
E(\delta\_k) = \int\_{f \le \delta\_k} f = \sum\_{j \ge k} \delta\_j \mu(f = \delta\_j)
\ge \delta\_k \mu(f = \delta\_k) = 2^{-k} = e(\delta\_k).
$$


