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?
Does exponential decrease also happen in continuous case?


No.  Feige and Verbitsky [FV02] showed that for every n, there is a game G (with finite sets of questions and answers) such that v(G)≤3/4 and v(Gn)≥1/8.  Because your formulation generalizes games with finite sets of questions and answers of any size, parallel repetition (of any finitely many times) cannot decrease the value of a game from 3/4 to 1/8.

[FV02] Uriel Feige and Oleg Verbitsky.  Error reduction by parallel repetition—A negative result.  Combinatorica, 22(4):461–478, Oct. 2002.  doi:10.1007/s00493-002-0001-0.