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  "body": "\n## Proof\n\nWe show that the half-lines form a π-system that generates the Borel sets, and then apply Dynkin's π-λ theorem.\n\n1. Let $\\mu$ and $\\nu$ be [[probability distributions|0QB913WM5E]] on $\\R$ with its [[Borel σ-algebra|0BVHZXWKCD]] $\\mathcal{B}(\\R)$. Assume $\\mu((-\\infty, x]) = \\nu((-\\infty, x])$ for all $x \\in \\R$.\n2. Let $\\mathcal{P} = \\set{(-\\infty, x] : x \\in \\R}$. It is nonempty, and $(-\\infty, x] \\cap (-\\infty, y] = (-\\infty, \\min(x, y)] \\in \\mathcal{P}$. So $\\mathcal{P}$ is a [[π-system|AQN6ZN66NK]].\n3. Let $\\sigma(\\mathcal{P})$ be the smallest [[σ-algebra|04WSW8T3SY]] on $\\R$ that contains $\\mathcal{P}$.\n4. Each $(-\\infty, x]$ is [[closed|0G5AV9Q4RR]], because its complement $(x, \\infty)$ is [[open|01ZQQDCERP]]. So $(-\\infty, x] \\in \\mathcal{B}(\\R)$, and hence $\\sigma(\\mathcal{P}) \\subseteq \\mathcal{B}(\\R)$.\n5. Let $a \u003c b$. Then $(a, b] = (-\\infty, b] \\cap (-\\infty, a]^c$ lies in $\\sigma(\\mathcal{P})$. The same formula with $b \\le a$ gives $\\varnothing \\in \\sigma(\\mathcal{P})$.\n6. We claim $(a, b) = \\bigcup_{n \\ge 1} (a, b - 1/n]$. Each set on the right lies in $(a, b)$. If $a \u003c x \u003c b$, the [[Archimedean property|0A3GWZ86AY]] gives $n$ with $n (b - x) \u003e 1$, so $x \\le b - 1/n$.\n7. By steps 5 and 6, every open interval $(a, b)$ with $a \u003c b$ lies in $\\sigma(\\mathcal{P})$.\n8. Let $U \\subseteq \\R$ be open. Let $J$ be the set of pairs $(q, r) \\in \\Q \\times \\Q$ with $q \u003c r$ and $(q, r) \\subseteq U$.\n9. The set $\\Q$ is [[countable|0DS9KWFDK2]], so $\\Q \\times \\Q$ is [[countable|08G4XH86JC]]. An injection of $\\Q \\times \\Q$ into $\\N$ restricts to $J$, so $J$ is [[countable|0ZG61TFBES]].\n10. We claim $U = \\bigcup_{(q, r) \\in J} (q, r)$. The inclusion $\\supseteq$ holds by the choice of $J$.\n11. For $\\subseteq$, let $x \\in U$. Since $U$ is open in $\\R$, there is $\\varepsilon \u003e 0$ with $(x - \\varepsilon, x + \\varepsilon) \\subseteq U$.\n12. By the [[density of the rationals|0STX0WRDNG]], there are rationals $q$ and $r$ with $x - \\varepsilon \u003c q \u003c x \u003c r \u003c x + \\varepsilon$. Then $(q, r) \\in J$ and $x \\in (q, r)$.\n13. By steps 7, 9 and 10, $U$ is a countable union of sets in $\\sigma(\\mathcal{P})$, or $U = \\varnothing$. So $U \\in \\sigma(\\mathcal{P})$.\n14. Thus $\\sigma(\\mathcal{P})$ is a σ-algebra that contains every open set, so $\\mathcal{B}(\\R) \\subseteq \\sigma(\\mathcal{P})$. With step 4, $\\sigma(\\mathcal{P}) = \\mathcal{B}(\\R)$.\n15. Let $\\mathcal{D} = \\set{B \\in \\mathcal{B}(\\R) : \\mu(B) = \\nu(B)}$. By the hypothesis, $\\mathcal{P} \\subseteq \\mathcal{D}$.\n16. The triples $(\\R, \\mathcal{B}(\\R), \\mu)$ and $(\\R, \\mathcal{B}(\\R), \\nu)$ are [[probability spaces|0ZYBZ34RT2]], because $\\mu$ and $\\nu$ are [[probability measures|0VGMEBVW1M]]. So the results on probability spaces apply to them.\n17. $\\R \\in \\mathcal{D}$, because $\\mu(\\R) = 1 = \\nu(\\R)$.\n18. Let $A, B \\in \\mathcal{D}$ with $A \\subseteq B$. By [[monotonicity of probability|0464Z0JP2E]], $\\mu(B \\setminus A) = \\mu(B) - \\mu(A) = \\nu(B) - \\nu(A) = \\nu(B \\setminus A)$. So $B \\setminus A \\in \\mathcal{D}$.\n19. Let $A_1 \\subseteq A_2 \\subseteq \\cdots$ be sets in $\\mathcal{D}$ with union $A$. By [[continuity from below|0X8HBKCSYB]], $\\mu(A) = \\lim_n \\mu(A_n) = \\lim_n \\nu(A_n) = \\nu(A)$. So $A \\in \\mathcal{D}$.\n20. By steps 17 to 19, $\\mathcal{D}$ is a [[Dynkin system|69460Z6DMW]] on $\\R$.\n21. By [[Dynkin's π-λ theorem|BBHM1A36M7]] and steps 2, 15 and 20, $\\sigma(\\mathcal{P}) \\subseteq \\mathcal{D}$.\n22. By step 14, $\\mathcal{B}(\\R) \\subseteq \\mathcal{D}$. So $\\mu(B) = \\nu(B)$ for each Borel set $B \\subseteq \\R$.\n\n## Idea\n\nThe sets on which two probability measures agree are closed under differences of nested sets and increasing unions, so they form a Dynkin system. The half-lines are closed under intersection and generate all Borel sets. Dynkin's π-λ theorem then spreads the agreement from the half-lines to all Borel sets.\n",
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