my setting is the following.

I have a sequence of zero sum games $G_n $ in which the strategy space is $[0,1]$, there are two players, and payoff functions are given by the continuous (no other restrictions so far) functions $U_i^n(x_1,x_2)$ for each player $i=1,2$.

Now, I know that the payoff functions $U_i^n$ converge pointwise to a (discontinuous) limit $U_i$. We can write down a correspondence $$EQ: G_n \rightarrow M([0,1])$$ that maps a game to its nash equilibrium. $M([0,1])$ are the probability measures on $[0,1]$.

I am struggling to understand the following. Help and appropriate references are much appreciated.

• Given the convergence of payoff functions, in what sense can I say the games $\lbrace G_n \rbrace $ converge to a limit game $G$?
• I have studied the limit game, and found it to have a unique (mixed strategy =proababilistic) equilibrium. What conditions do I need on the correspondence so that some sequence of equilibria of $\lbrace G_n \rbrace$ converge (uniformly) to the equilibrium of the limit game?

Any help or references are greatly appreciated.