# What are exactly the information given by the CDF of a random variable?

Let $$(\Omega, \mathcal{F}, \mathbb{P})$$ be a probability space and $$X : (\Omega, \mathcal{F}) \rightarrow (\mathbb{R}, \mathcal{F}_E)$$ a random variable that admits a density function $$f_X : \mathbb{R} \rightarrow \mathbb{R}$$.

I have some questions about the information given by the cumulative distribution function of $$X$$ ($$F_X : \mathbb{R} \rightarrow [0,1]$$).

i) Does the CDF determine $$f_X$$ 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 $$F_X(t) = \int_{- \infty}^{t}f_X(x)dx$$, 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 $$\forall x \in \mathbb{R}$$, $$\mathbb{P}(X = x) = F_X(x) - F_X(x-)$$ and $$\mathbb{P}(X \leq x) = F_X(x)$$, 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 $$\{X \in (a,b): a, b \in \mathbb{R}\}$$ but I'm not quite sure about that, and particularly regarding the uniqueness.

By subtraction, the CDF $$F(x) := P(X \leq x)$$ determines the probabilities $$P(X \in [a, b])$$ for every $$a, b \in \mathbb{R}$$. By an approximation theorem, such as Caratheodory's theorem, this information determines $$P(X \in E)$$ for every $$E \in B(\mathbb{R})$$. So (ii) is true.