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  "body": "\n## Definition\n\nLet $(X, \\Sigma)$ be a [[measurable space|N4KDT83V3S]].\nA *simple function* on $X$ is a [[measurable function|YYNCJSR0PJ]] $s \\colon X \\to \\R$ whose image $s(X)$ is a finite set.\n\n## Remarks\n\nEquivalently, $s = \\sum_{k=1}^{n} a_k \\mathbf{1}_{A_k}$ for some $n \\ge 0$, real numbers $a_k$ and sets $A_k \\in \\Sigma$, where $\\mathbf{1}_{A_k}$ is the [[indicator function|0RCQZ36MME]] of $A_k$.\nThe sets $A_k$ can be chosen pairwise disjoint, with distinct values $a_k$.\nFor a measure $\\mu$ and $a_k \\ge 0$, the integral is $\\int s \\, d\\mu = \\sum_k a_k \\mu(A_k)$, with $0 \\cdot \\infty = 0$.\nExample: on $\\R$ with the Borel sets, $s = 3 \\cdot \\mathbf{1}_{[0, 1]} - \\mathbf{1}_{\\Q}$ is simple; its values are $-1$, $0$, $2$ and $3$.\nNon-example: $x \\mapsto x$ on $[0, 1]$ is measurable but has infinitely many values.\nEvery measurable $f \\colon X \\to [0, \\infty]$ is the pointwise limit of a non-decreasing sequence of nonnegative simple functions.\nSome authors drop the measurability condition, or allow complex values.\n",
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