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  "body": "\n## Definition\n\nLet $X$ be a [[metric space|063W5JMY85]], for example a subset of $\\R$.\nLet $\\omega_1$ be the first uncountable [[ordinal|0RDJK7EQYY]].\nFor each ordinal $\\alpha \u003c \\omega_1$, define a set $B_\\alpha$ of functions $X \\to \\R$ as follows.\n\n1. $B_0$ is the set of [[continuous functions|0P27W9RW0G]] $X \\to \\R$.\n2. For $0 \u003c \\alpha \u003c \\omega_1$, $B_\\alpha$ is the set of functions $f \\colon X \\to \\R$ that are [[pointwise limits|056FSKG32J]] $f(x) = \\lim_{n \\to \\infty} f_n(x)$ of sequences $(f_n)$ in $\\bigcup_{\\beta \u003c \\alpha} B_\\beta$.\n\nA function of *Baire class* $\\alpha$ is an element of $B_\\alpha$.\nA *Baire function* is a function that lies in $B_\\alpha$ for some $\\alpha \u003c \\omega_1$.\n\n## Remarks\n\nThe classes increase: $B_\\beta \\subseteq B_\\alpha$ for $\\beta \\le \\alpha$, since a function is the limit of a constant sequence.\nThe set of Baire functions is the smallest set of functions $X \\to \\R$ that contains the continuous functions and is closed under pointwise limits of sequences.\nFor $X = \\R$, the Baire functions are exactly the functions that are measurable for the [[Borel sigma-algebra|0BVHZXWKCD]].\n\nExample: if $F \\colon \\R \\to \\R$ is differentiable, then $F'$ is of Baire class $1$.\nIndeed $F'(x) = \\lim_{n \\to \\infty} n \\big(F(x + 1/n) - F(x)\\big)$, and each function on the right is continuous.\n\nNon-example: the [[Dirichlet function|0T4EJ8ZJM9]] $\\mathbf 1_\\Q$ is not of Baire class $1$.\nA function of Baire class $1$ on $\\R$ is continuous at the points of a dense set, while $\\mathbf 1_\\Q$ is continuous nowhere.\nIt is of Baire class $2$, since $\\mathbf 1_\\Q(x) = \\lim_{m \\to \\infty} \\lim_{n \\to \\infty} \\cos(m! \\pi x)^{2n}$.\n",
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