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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]].\nLet $F_n \\colon X \\to [-\\infty, +\\infty]$ for $n \\in \\N$ and $F \\colon X \\to [-\\infty, +\\infty]$ be functions with values in the [[extended real line|0GSZ7B0CGR]].\nThe [[sequence|0G9EKM0E15]] $(F_n)$ *Γ-converges* to $F$ if the following two conditions hold for every $x \\in X$.\n\n1. (Lower bound.) For every sequence $(x_n)$ in $X$ with $x_n \\to x$,\n$$F(x) \\le \\liminf_{n \\to \\infty} F_n(x_n).$$\n2. (Recovery sequence.) There is a sequence $(x_n)$ in $X$ with $x_n \\to x$ and\n$$F(x) \\ge \\limsup_{n \\to \\infty} F_n(x_n).$$\n\nThe function $F$ is then called the *Γ-limit* of $(F_n)$, written $F = \\Gamma\\text{-}\\lim_n F_n$.\nHere $\\liminf$ is the [[limit inferior|033RZ32T4Q]] and $\\limsup$ is the [[limit superior|0T3RZQXDYD]].\n\n## Remarks\n\nThe two conditions together give $F(x) = \\lim_n F_n(x_n)$ along the recovery sequence.\nA Γ-limit is unique when it exists, and it is always lower semicontinuous.\nFor a family $(F_\\varepsilon)_{\\varepsilon \u003e 0}$, one says that $F_\\varepsilon$ Γ-converges to $F$ as $\\varepsilon \\to 0$ if $F_{\\varepsilon_k}$ Γ-converges to $F$ for every sequence $\\varepsilon_k \\to 0$.\nThe same definition with neighbourhoods instead of sequences works in any topological space, and it agrees with this one in metric spaces.\n\nExample: $F_n(x) = \\sin(nx)$ on $\\R$ Γ-converges to the constant $F = -1$.\nThe lower bound holds because $\\sin(n x_n) \\ge -1$ for every $n$.\nFor a recovery sequence, pick $x_n$ with $\\abs{x_n - x} \\le 2\\pi/n$ and $\\sin(n x_n) = -1$; then $x_n \\to x$ and $F_n(x_n) = -1$.\n\nNon-example: let $F(0) = 1$ and $F(x) = 0$ for $x \\ne 0$, and let $F_n = F$ for all $n$.\nThe constant sequence does not Γ-converge to $F$: for $x_n = 1/n \\to 0$ one has $\\liminf_n F_n(x_n) = 0 \u003c 1 = F(0)$, so the lower bound fails at $0$.\nIts Γ-limit is the function $0$, the lower semicontinuous envelope of $F$.\n",
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