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  "body": "\n## Statement\n\nLet $X$ be a [[metric space|0VKD1TP7F3]] and let $\\mu, \\mu_1, \\mu_2, \\dots$ be Borel [[probability measures|059TJN0NZ1]] on $X$.\nFor a set $A \\subseteq X$, let $\\partial A$ be its boundary.\nThe following conditions are equivalent.\n\n1. $\\mu_n$ [[converges weakly|MBW9JCGZC1]] to $\\mu$: $\\int f \\, d\\mu_n \\to \\int f \\, d\\mu$ for every bounded continuous $f \\colon X \\to \\R$.\n2. $\\int f \\, d\\mu_n \\to \\int f \\, d\\mu$ for every bounded [[Lipschitz|0AW1E6YTJY]] function $f \\colon X \\to \\R$.\n3. $\\limsup_{n \\to \\infty} \\mu_n(F) \\le \\mu(F)$ for every [[closed set|0ZCK8DQYN1]] $F \\subseteq X$.\n4. $\\liminf_{n \\to \\infty} \\mu_n(G) \\ge \\mu(G)$ for every [[open set|02BH3MCBQB]] $G \\subseteq X$.\n5. $\\mu_n(A) \\to \\mu(A)$ for every Borel set $A$ with $\\mu(\\partial A) = 0$.\n\n## Remarks\n\nCondition 1 implies 2 at once.\nFor 2 implies 3, the Lipschitz functions $f_k(x) = \\max(0, 1 - k \\, d(x, F))$ decrease to the indicator of $F$ as $k \\to \\infty$.\nConditions 3 and 4 are equivalent by taking complements.\nConditions 3 and 4 applied to the closure and the interior of $A$ give 5.\nFor 5 implies 1, write $\\int f \\, d\\mu$ as an integral of $\\mu(\\set{f \u003e t})$ over $t$; for all but countably many $t$ the set $\\set{f \u003e t}$ has a $\\mu$-null boundary.\n\nExample: $\\delta_{1/n}$ converges weakly to $\\delta_0$ on $\\R$.\nFor the open set $G = (0, 1)$, one has $\\delta_{1/n}(G) = 1$ for $n \\ge 2$ and $\\delta_0(G) = 0$, so the inequality in 4 can be strict.\nFor $A = \\set{0}$, the boundary is $\\set{0}$, which has $\\delta_0$-measure $1$.\nSo condition 5 makes no claim, and indeed $\\delta_{1/n}(A) = 0$ does not tend to $\\delta_0(A) = 1$.\n\n## Explanations\n\n### L1 human\n\nThe theorem lists several equivalent ways to say that measures converge weakly.\nIn the limit, a closed set can gain mass and an open set can lose mass, but not the other way round.\nSets whose boundary carries no limit mass have converging measures.\n\n### L2 human\n\nOne uses whichever form is easiest to check.\nCondition 2 is handy because Lipschitz functions are easy to control, while condition 5 gives convergence of probabilities of concrete events.\nFor real random variables, condition 5 applied to half-lines gives convergence of distribution functions at continuity points.\nThe theorem is a standard tool together with Prokhorov's theorem and the continuous mapping theorem.\n",
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