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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.

Thanks in advance for your help.

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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.

(i) follows (though, as always, the density is unique only up to almost everywhere equality).

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