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  "body": "\n## Summary\n\nA sequence of [[probability measures|059TJN0NZ1]] on a [[metric space|0VKD1TP7F3]] converges weakly when the integrals of every bounded continuous function converge.\nIt is the mode of convergence behind [[convergence in distribution|0SRYJGXKR4]] of random variables and the central limit theorem.\nMass may move a little in the limit, so the measure of a fixed set need not converge.\n\n## Explanations\n\n### L1 human\n\nTwo measures are close in this sense if they look alike to every smooth test, even if they put mass on slightly different points.\nPicture a point mass at $1/n$ sliding towards $0$.\nFor every continuous bounded $f$, the value $f(1/n)$ tends to $f(0)$, so these point masses converge weakly to the point mass at $0$.\nYet the set $\\set{0}$ has mass $0$ for every $n$ and mass $1$ in the limit.\nWeak convergence only sees the overall shape, not exact sets.\n",
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