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  "body": "\n## Definition\n\nLet $X$ be a [[metric space|0VKD1TP7F3]] with its [[Borel σ-algebra|0WQXY32Q8D]].\nLet $\\mu$ and $\\mu_1, \\mu_2, \\dots$ be [[probability measures|059TJN0NZ1]] on $X$.\nThe sequence $(\\mu_n)$ *converges weakly* to $\\mu$ if\n$$\\lim_{n \\to \\infty} \\int_X f \\, d\\mu_n = \\int_X f \\, d\\mu$$\nfor every bounded [[continuous function|0X3WGX95FE]] $f \\colon X \\to \\R$.\nThe integrals are [[Lebesgue integrals|DWJF7ABPR1]].\n\n## Remarks\n\nCommon notations are $\\mu_n \\Rightarrow \\mu$ and $\\mu_n \\xrightarrow{w} \\mu$.\nSome authors call the same notion narrow convergence.\nWeak limits are unique on a metric space.\n\nExample: let $\\delta_a$ be the [[Dirac measure|0ACRGYNHSR]] at $a$ on $\\R$.\nThen $\\delta_{1/n}$ converges weakly to $\\delta_0$.\nIndeed $\\int f \\, d\\delta_{1/n} = f(1/n)$, and $f(1/n) \\to f(0)$ because $f$ is continuous.\n\nNon-example: $\\delta_n$ does not converge weakly to $\\delta_0$.\nTake $f(x) = 1/(1 + x^2)$, which is bounded and continuous.\nThen $\\int f \\, d\\delta_n = 1/(1 + n^2) \\to 0$, but $\\int f \\, d\\delta_0 = 1$.\n\nRandom variables converge in distribution exactly when their laws converge weakly.\n\n## Explanations\n\n### L1 human\n\nWeak convergence asks that every bounded continuous average converges.\nIt allows mass to shift slightly, which makes it the natural notion for limit theorems in probability.\n",
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