Stochastic Processes – Is the probability space really an important part of a weak solution to an SDE?

A weak solution is a tuple $ ( Omega, mathcal F, mathbb P, X, B, ( mathcal F_t)) $ with $ ( Omega, mathcal F, mathbb P) $ a probability space, $ B $ a $ ( mathcal F_t) $Brownian motion and $ X $ a $ ( mathcal F_t) $– adapted process satisfying certain regularity conditions and of course the given SDE.

In the two examples that I saw, we always chose a probability space $ ( Omega, mathcal F, mathbb P) $ which is rich enough to admit Brownian motion.

So, can not we just use the same space each time?