Random Lipschitz constant in a finite net bound
= Random Lipschitz constant in a finite net bound
Suppose $|Z(u)-Z(v)|\leq L\|u-v\|$ with a random finite $L$. On $L\leq\ell$, a <finite net> with radius $\eta$ bounds the full <supremum> by the grid maximum plus $\ell\eta$. The <Markov inequality> controls the exceptional event using $\mathbb EL$, and a <union bound> controls the grid maximum. Choosing the net spacing balances these two costs.