# Is there a continuous version of parallel repetition theorem

Raz's Parallel pretition theorem is an important result in PCP, inapproximation, etc. The theorem is fomalized as follows.

A game $G=(\mathcal{S},\mathcal{T},\mathcal{A},\mathcal{B},\pi, V)$, where $\mathcal{S},\mathcal{T},\mathcal{A},\mathcal{B}$ are finite sets, $\pi$ is a distribution on $\mathcal{S}\times\mathcal{T}$, and predicate $V:\mathcal{S}\times\mathcal{T}\times\mathcal{A}\times\mathcal{B}\rightarrow\{0,1\}$. Define value of the game $$v(G)=\max_{h_A\in\mathcal{H}_A,h_B\in\mathcal{H}_B}\sum_{s,t}\pi(s,t)V(s,t,h_A(s),h_B(t))$$ And $n$-fold game $G^n=(\mathcal{S}^n,\mathcal{T}^n,\mathcal{A}^n,\mathcal{B}^n,\pi^n, V^n)$. The theorem says if $v(G)\leq 1-\epsilon,$ then $v(G^n)\leq (1-\epsilon^c)^{\Omega(\frac{n}{\log\max\{|A|,|B|\}})}$.

My quesion is what happen if the sets are infinite, in a continuous space. Say if $\mathcal{S},\mathcal{T},\mathcal{A},\mathcal{B}$ are subsets of a space, say $R^n$, or more abstract spaces. All the rest are same. Raz's theorem only gives a trivial upper bound $1$ since the sizes of answer sets are infinite. Obviously $n$-fold value is upper bounded by single copy. Does exponential decrease also happen in continuous case? Would it be more interesting to restrict $\mathcal{H}_A,\mathcal{H}_B$ to be collections of continuous functions or $C^{\infty}$ functions or measureable functions?