Sup-Inf Choice Function Identity

Let $f:S\times T\to\mathbb R$, where $S$ and $T$ are arbitrary nonempty sets. Let $B$ be the set of all maps $\beta:S\to T$. Then $$ \sup_{s\in S}\inf_{t\in T} f(s,t) = \inf_{\beta\in B}\sup_{s\in S} f(s,\beta(s)). $$

References

• [MFoGT, Section 3.5, Exercise 5] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Sup-inf identity using choice functions from S to T.