Uniform Hashing. Understanding space occupancy and choice of functions
Suppose that we sample a hash function $h$ at random from a collection $\mathcal{H}\subseteq [m]^U$ (we denote by $[m]$ the set $\{0,1,...,m-1\}$). The best possible thing we can require from this choice is that $\left(h(x_1),h(x_2),....,h(x_{|u|})\right)$ are jointly uniformly distributed over $[m]^U$, where $U=\{x_1,...,x_{|U|}\}$ (this in turn implies independence). Fix some $h_0\in [m]^U$, then under the previous requirement we have:
$\Pr\limits_{h\sim\mathcal{H}}[h=h_0]=\Pr\limits_{h\sim\mathcal{H}}\left[h(x_1)=h_0(x_1)\land...\land h(x_{|U|})=h_0(x_{|U|})\right]=\frac{1}{m^{|U|}}$.
It follows that $h$ is uniformly distributed over all possible functions in $[m]^U$ (in any other case we will fail either in uniformity or in independence). This is what is meant by &quot;$h$ should be anyone of these functions&quot;.