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tl;dr: informally: You cannot know the value of the difference of two random variables by knowing their sum. Consider the following set $A =\{ \omega \in \Omega | W_1(t)(\omega) - W_2(t)(\omega) \in [0,1] \}$ This set is of course in the Filtration generated by $W_1$ and $W_2$ since the addition of measurable functions is measurable. Is this set in the (...

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