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  "body": "\n## Example\n\nLet $X$ be a set and let $\\Sigma$ be its power set.\nFor $A \\subseteq X$, let $\\mu(A)$ be the number of elements of $A$ if $A$ is finite, and $\\mu(A) = \\infty$ if $A$ is infinite.\nThen $\\mu$ is a [[measure|PR11BZQQPG]] on $(X, \\Sigma)$, the counting measure.\n\nFor the check, $\\mu(\\varnothing) = 0$.\nLet $A_1, A_2, \\ldots$ be pairwise disjoint subsets of $X$ with union $A$.\nSuppose all $A_n$ are finite and only finitely many are nonempty.\nThen $A$ is finite, and its number of elements is the sum of those of the $A_n$.\nOtherwise $A$ is infinite, and the sum $\\sum_n \\mu(A_n)$ is $\\infty$ too, since it has a term $\\infty$ or infinitely many terms at least $1$.\n\n## Remarks\n\nCounting measure is [[σ-finite|WEBBT18D4X]] exactly when $X$ is countable.\nOn $X = \\N$, the integral of a non-negative function $f$ with respect to counting measure is the sum $\\sum_{n} f(n)$.\nSo series of non-negative terms are a special case of Lebesgue integrals.\n\n## Explanations\n\n### L1 human\n\nCounting measure gives each set its number of elements.\nThe set $\\set{2, 5, 7}$ has measure $3$, and the set of even numbers has measure $\\infty$.\nIt is the simplest measure, and it turns sums into integrals.\n",
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