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  "body": "\n## Definition\n\nLet $\\mathcal{X}$ be a [[topological space|0CQHFPGJ45]].\nA *rate function* on $\\mathcal{X}$ is a function $I \\colon \\mathcal{X} \\to [0, \\infty]$ that is [[lower semicontinuous|RN4G8SGP6P]].\nEquivalently, for every $a \\ge 0$, the sublevel set $\\set{x \\in \\mathcal{X} : I(x) \\le a}$ is [[closed|0ZCK8DQYN1]].\nA rate function $I$ is *good* if every sublevel set $\\set{x : I(x) \\le a}$, for $a \\ge 0$, is [[compact|0TVJZYB6SH]].\n\n## Remarks\n\nExample: $I(x) = x^2/2$ on $\\R$ is a good rate function.\nIt is continuous, so lower semicontinuous, and $\\set{I \\le a} = [-\\sqrt{2a}, \\sqrt{2a}]$ is compact.\n\nExample: on $\\R$, let $I(x) = x \\log x + (1 - x) \\log(1 - x) + \\log 2$ for $x \\in [0, 1]$, with $0 \\log 0 = 0$, and $I(x) = \\infty$ otherwise.\nIt is a good rate function, the one for fair coin tossing in [[Cramér's theorem|RZZK51CXMN]].\n\nNon-example: on $\\R$, let $I(0) = 1$ and $I(x) = 0$ for $x \\neq 0$.\nThen $\\set{I \\le 0} = \\R \\setminus \\set{0}$ is not closed, so $I$ is not lower semicontinuous.\n\nNon-example of goodness: $I = 0$ on $\\R$ is a rate function, but $\\set{I \\le 0} = \\R$ is not compact.\n\nIn a large deviation principle, the rate function is unique when $\\mathcal{X}$ is a regular topological space, for example a metric space.\nA good rate function attains its infimum on every non-empty closed set.\n",
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