Ville Theorem By Discretization
Let $X=Y=[0,1]$ and let $f:X\times Y\to\mathbb R$ be continuous. For each $n\ge1$, form the finite grid $$ X_n=Y_n=\{0,1,\ldots,2^n\} $$ and the finite matrix game $$ G_n(i,j)=f(i/2^n,j/2^n). $$ Let $v_n$ be the mixed value of $G_n$.
Proof
The proof of Ville's theorem has two steps.
First, use uniform continuity of $f$ to transfer guarantees from sufficiently fine grid games to the original continuous game. If player 1 plays an optimal mixed strategy on a fine grid, interpreted as a finitely supported Borel probability measure on $[0,1]$, then player 1 guarantees $\limsup_n v_n$ up to any prescribed $\varepsilon>0$. Dually, player 2 guarantees $\liminf_n v_n$ from above. These two inequalities force existence of a value.
Second, regard optimal grid strategies as Borel probability measures on the compact interval $[0,1]$. By compactness of probability measures in the weak topology, extract weakly convergent subsequences. Continuity of $f$ lets the payoff functional pass to the limit, and the limiting measures are optimal strategies. The finite grid strategies also give the finite-support $\varepsilon$-optimal strategies.
References
- [MFoGT, Section 2.8, Exercise 3] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Proof of Ville's theorem by finite discretizations and weak convergence.