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I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.

I have a question about a step in theorem 3.8 on page 17. Let me give the set-up. Each player $i$ has the payoff function $$\mathcal{J^i}(\alpha^1, \ldots \alpha^N) = \mathbb{E}\left[\int_0^T \frac12|\alpha_s^i|^2 + F\left(X_s^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right) \ ds + G\left(X_T^i, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_T^j}\right)\right].$$

The dynamics of the player are given by $dX_t^i = \alpha_t^i \ dt + \sqrt{2}\ dB_t^i$, as usual. $F$ and $G$ are Lipschitz in both variables.

The $X_t^i$ are i.i.d. with law $m(t)$. The crucial step that I fail to understand is the following: there is $N_0$ large such that for $N \ge N_0$, $$\mathbb{E}\left[\sup_{|y|\le 1/\sqrt{\epsilon}} \left|F(y, m(t)) - F\left(y, \frac{1}{N-1}\sum_{j\neq i} \delta_{X_s^j}\right)\right|\right]\le \epsilon$$ for any $t \in [0,T]$. Note that the subscript on the $X^i$ is $s$, not $t$.

He says this is an application of Hewitt-Savage. However, I fail to see how this works at all. I'm not sure where to introduce my integrals, and I definitely don't see what role the restriction $|y|\le 1/\sqrt{\epsilon}$ is playing here. $F$ is uniformly Lipschitz in the $m$ variable, independent of the parameter $y$. Because this is the case, clearly there has to be something more do just passing to the Lipschitz bound on that difference, because we'd be dropping anything to do with $y$ (and then I really can't see how Hewitt-Savage would help). Can someone help shed light on this step?

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From the reference, the limit under $n \rightarrow + \infty$ in the result of corollary 5.13 (https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf) implies that there exists $1 \leq N_0$ and $0 \leq \epsilon_1$ such that for all $N_0 \leq N$, $$\mathbb{E}\left[ \underset{y}{\sup} \left| F \left( y, \frac{1}{N - 1} \underset{n \geq 2}{\sum} \delta_{\bar{X}^j_s} \right) - F(y, m(s)) \right| \right] \leq \epsilon_1$$ where the expectation indicates an integral over the joint probability distribution of the states $(\bar{X}^j_s)$. The Lipschtiz condition w.r.t. $m(\cdot)$ implies that there exists $0 < \epsilon_2$ such that for given $0 \leq s, t$, $$\mathbb{E}\left[ \underset{y}{\sup} \left| F \left(y,m(t) \right) - F(y, m(s)) \right| \right] \leq \epsilon_2.$$ Applying the triangle inequality and choosing $\epsilon = \epsilon_1 + \epsilon_2$ we can obtain the inequality which takes the $\sup$ over all $y$, so that it is also true for $|y| \leq 1/\sqrt{\epsilon}$. The inequality constraint referred to in the question is related to the construction of the proof and not directly related to showing this particular expectation-inequality per se.

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