I have some questions about the information given by the cumulative distribution function of X ().
i) Does the CDF determine uniquely ?
ii) Does the CDF determine the law of X uniquely ?
iii) Can we find a subset of events that uniquely determine the CDF ?
For i), I would say that as , the density function is the derivative of the CDF and therefore the CDF can only define one density function. But I feel like maybe we could find a counterexample by considering Lebesgue measure and taking sets of the form (a,b) and [a,b].
For ii), as , and , I am tempted to say that the law of X is indeed by definition determined uniquely by the CDF.
For iii), maybe we could take all events of the form but I'm not quite sure about that, and particularly regarding the uniqueness.