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  "body": "\n## Definition\n\nLet $2^\\omega$ be [[Cantor space|W9VYJRBF9N]], the set of infinite binary sequences.\nFor a finite binary string $\\sigma$ of length $\\abs{\\sigma}$, let $[\\sigma] \\subseteq 2^\\omega$ be the set of sequences that begin with $\\sigma$.\nLet $\\mu$ be the uniform measure on $2^\\omega$, with $\\mu([\\sigma]) = 2^{-\\abs{\\sigma}}$.\nA *Martin-Löf test* is a sequence $(U_n)_{n \\in \\N}$ of subsets of $2^\\omega$ such that:\n\n1. there is a [[computably enumerable|TN9G24VABN]] set $W$ of pairs $(n, \\sigma)$ with $U_n = \\bigcup \\set{[\\sigma] : (n, \\sigma) \\in W}$ for every $n$;\n2. $\\mu(U_n) \\le 2^{-n}$ for every $n$.\n\nA sequence $X \\in 2^\\omega$ is *Martin-Löf random*, or *algorithmically random*, if $X \\notin \\bigcap_n U_n$ for every Martin-Löf test $(U_n)$.\n\n## Remarks\n\nExample: the binary expansion of Chaitin's halting probability $\\Omega$ is Martin-Löf random.\nAlso, the set of non-random sequences has measure $0$, because there is a universal Martin-Löf test whose intersection contains every non-random sequence.\n\nNon-example: a computable sequence $X$ is not random.\nLet $X \\upharpoonright n$ be the string of the first $n$ bits of $X$.\nThe sets $U_n = [X \\upharpoonright n]$ form a Martin-Löf test: they are uniformly computable and $\\mu(U_n) = 2^{-n}$.\nAnd $X$ lies in all of them.\n\nThe Levin–Schnorr theorem gives an equivalent form with [[Kolmogorov complexity|8PY3K6NCKF]].\nLet $K$ be prefix-free complexity.\nThen $X$ is Martin-Löf random if and only if there is a constant $c$ with $K(X \\upharpoonright n) \\ge n - c$ for all $n$.\n",
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