Is there a continuous version of parallel repetition theorem
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.