According to the definition, for a Brownian motion it holds that
$W_0 = 0$,
$W_t - W_s \in N(0, t-s), \quad t > s$.
This implies that $W_t \in N(0, t)$, for all $t \geq 0$. Hence, the definition gives us the distribution of every single value in the process, if I'm not misunderstanding something. Wouldn't that mean that the definition uniquely defines the probability space (including the measure) for the process? Then, how can there be different Brownian motions under different measures?
I have read the answer to What is a Brownian motion "under the risk-neutral measure"?, but I still don't understand this. I am new to the subject, and do not know a lot about measure theory, so if it's possible that someone could give a somewhat simple explanation, perhaps in plain English, it would be very helpful.