In the study of zero sum games, Glicksberg's theorem (also Glicksberg's existence theorem) is a result that shows certain games have a minimax value. If A and B are Hausdorff compact spaces, and K is an upper semicontinuous or lower semicontinuous function on A × B {\displaystyle A\times B} , then
sup f inf g ∬ K d f d g = inf g sup f ∬ K d f d g {\displaystyle \sup _{f}\inf _{g}\iint K\,df\,dg=\inf _{g}\sup _{f}\iint K\,df\,dg}
where f and g run over Borel probability measures on A and B. The theorem is useful if f and g are interpreted as mixed strategies of two players in the context of a continuous game. If the payoff function K is upper semicontinuous, then the game has a value. The continuity condition may not be dropped: see example of a game with no value.
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