Abstract: In this paper, we give a novel proof for the following equality
$\mathop {\lim }\limits_{\alpha \to {0^ + }} {\alpha ^P}{d_f}(\alpha ) = \mathop {\lim }\limits_{\alpha \to \infty } {\alpha ^P}{d_f}(\alpha ) = 0$
for f∈L^p,q(X,μ) with 0<p<∞, and 0<q<∞.We also prove that the function α^p can not be improved for some sense. When q=∞, the above equality does not hold.

Key words: limiting behavior, distribution functions, Lorentz spaces