  In this paper, we first prove an interpolation inequality of Ehrling-type,
which is an improvement of a special case to the well known
Gargliardo-Nirenberg inequality. Then we apply it to study the classical
Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta
u-\nabla\cdot(u \nabla v), \\[6pt] \displaystyle v_t=\Delta v-v+u, \end{array}
\right. \end{equation} in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N\ge
2$) with smooth boundary. It is known that for any $\delta>0$, if $\int_\Omega
u^{\frac N2+\delta}(\cdot,t)$ is bounded, then the solution is global and
bounded. Here we show that the same conclusion holds for a weaker assumption:
the equi-integrability of $\{\int_\Omega u^\frac
N2(\cdot,t)|~t\in(0,T_{\max})\}$ can prevent blow up.
