Average integration is increasing with measure

Question

I was studying Evans' PDE, Morrey's Inequality. A step confused me: Is it correct that there is a constant depending only on $n$ so that $ \frac{\int_U f}{|U|}\leq C\frac{\int_V f}{|V|}$ if $U\subset V\subset \mathbb R^n$ and $f\geq 0$ ? One may assume U,V bounded and open. I had tried some ways such as "$\int_V f\cdot(|V|1_{\chi_U}) \underset{?}{\leq} |U| \int_V f $" by Holder's inequality but failed. Can anyone tell the trick ?