The article shows that in games with a measure space of players and discontinuous payoffs, there exists a pure strategy Nash equilibrium. This means that in certain complex games, players can reach a stable outcome where no one can improve their position by changing their strategy. The researchers extended previous work on continuous security to these types of games and applied their findings to optimal income tax problems. This shows that in scenarios like setting income tax rates, there is an optimal solution that can be reached through game theory.