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  "body": "\n## Statement\n\nLet $\\mu$ and $\\nu$ be [[probability distributions|0QB913WM5E]] on $\\mathbb{R}$ with its Borel σ-algebra.\nIf $\\mu((-\\infty, x]) = \\nu((-\\infty, x])$ for each $x \\in \\mathbb{R}$, then $\\mu(B) = \\nu(B)$ for each Borel set $B \\subseteq \\mathbb{R}$.\n\n## Remarks\n\nIn other words, two real random variables with the same [[cumulative distribution function|0MXMS3AE13]] are identically distributed.\nThe half-lines $(-\\infty, x]$ are closed under finite intersections and generate the [[Borel σ-algebra|0BVHZXWKCD]]. The proof uses Dynkin's π-λ theorem to pass from these half-lines to all Borel sets.\nSo to describe a real distribution, it is enough to give its cumulative distribution function.\n\n## Explanations\n\n### L1 human\n\nIf you know $P(X \\le x)$ for every $x$, then you know the probability of every reasonable event about $X$.\nFor example, $P(a \u003c X \\le b)$ is a difference of two values of the function, and more complex sets come from such intervals.\n",
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