Extending to a probability measure from a CDF For any real random variable $X$ it is sufficient to describe the cumulative distribution function to give the entire distribution. Why is that?
In other words, if we assign numbers on the collection of sets of the form $(-\infty, x]$ such that the assignment meets the conditions of the CDF (non-decreasing, right continuity, $F(-\infty)=0,F(+\infty)=1$)  then we can extend this assignment to all the Borel sets. How is that accomplished exactly?
I thought Cartheodory's theorem would be used here, but the statement of the theorem requires a ring of sets which $(-\infty, x]$ is not.
 A: You asked two different questions.

The first question was: For any real random variable $X$ it is sufficient to describe the cumulative distribution function to give the entire distribution. Why is that?
Two probability measures $P$ and $Q$ on $(\mathbb{R},\mathcal{B})$ that coincide on a generator of $\mathcal{B}$ that is closed to finite intersections, are identical (this is due to Dynkin's $\pi$-$\lambda$ theorem). The set $\big\{(-\infty,x] : x \in \mathbb{R}\big\}$ is such a generator.
Let $X$ and $Y$ be random variables with cdfs $F_X$ and $F_Y$, and with probability distributions $P_X$ and $P_Y$, respectively. If $F_X = F_Y$, then $P_X\big((-\infty,x]\big) = F_X(x) = F_Y(x) = P_Y\big((-\infty,x]\big)$ for every $x \in \mathbb{R}$, hence, $P_X = P_Y$, which is to say $X$ and $Y$ have the same distribution.

As for your second question: if we assign numbers on the collection of sets of the form $(-\infty, x]$ such that the assignment meets the conditions of the CDF (non-decreasing, right continuity, $F(-\infty)=0,F(+\infty)=1$)  then we can extend this assignment to all the Borel sets. How is that accomplished exactly?
It can be shown that if $F:\mathbb{R}\rightarrow[0,1]$ satisfies the axioms of a CDF, and we define $X:(0,1)\rightarrow\mathbb{R}$ to be the inverse of $F$ (more precisely $F$'s quantile function, which reduces to $F$'s inverse if $F$ is invertible) , then $F_X = F$, when $X$ is considered as a random variable over the probability space $\big((0,1), \mathcal{B}_{(0,1)}, \lambda\big|_{(0,1)}\big)$, $\lambda$ being the Lebesgue measure on the real line. $X$'s distribution $P_X$ is an extension of $F_X$ to all Borel sets.
