The cumulative distribution function determines the distribution

Statement

Let $\mu$ and $\nu$ be probability distributions on $\mathbb{R}$ with its Borel σ-algebra. If $\mu((-\infty, x]) = \nu((-\infty, x])$ for each $x \in \mathbb{R}$, then $\mu(B) = \nu(B)$ for each Borel set $B \subseteq \mathbb{R}$.

Remarks

In other words, two real random variables with the same cumulative distribution function are identically distributed. The half-lines $(-\infty, x]$ are closed under finite intersections and generate the Borel σ-algebra. The proof uses Dynkin's π-λ theorem to pass from these half-lines to all Borel sets. So to describe a real distribution, it is enough to give its cumulative distribution function.

Explanations

L1 human

If you know $P(X \le x)$ for every $x$, then you know the probability of every reasonable event about $X$. For example, $P(a < X \le b)$ is a difference of two values of the function, and more complex sets come from such intervals.