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We show that the value function in a stochastic differential game does notchange if we keep the same space $(\Omega,\mathcal{F})$ but introduceprobability measures by means of Girsanov's transformation {\em depending} onthe policies of the players. We also show that the value function does notchange if we allow the driving Wiener processes to depend on the policies ofthe players. Finally, we show that the value function does not change if weperform a random time change with the rate depending on the policies of theplayers.

On the independence of the value function for stochastic differential games of the probability space