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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $B = [a_1, b_1] \\times \\cdots \\times [a_n, b_n]$ be a closed box in $\\R^n$ with $a_i \u003c b_i$.\nIts volume is $\\abs{B} = (b_1 - a_1) \\cdots (b_n - a_n)$.\nA partition $P$ of $B$ is given by a partition of each interval $[a_i, b_i]$; it cuts $B$ into finitely many subboxes $B_1, \\dots, B_N$.\nThe mesh of $P$ is the largest diameter of the $B_j$.\nA tagged partition also chooses a point $\\xi_j \\in B_j$ for each $j$.\nFor a function $f \\colon B \\to \\R$, the [[Riemann sum|0JAEHXTVV5]] of a tagged partition is $\\sum_{j=1}^N f(\\xi_j) \\abs{B_j}$.\n\nThe function $f$ is *Riemann integrable* on $B$ if there is a real number $I$ with the following property.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that every tagged partition of mesh less than $\\delta$ has a Riemann sum within $\\varepsilon$ of $I$.\nThen $I$ is the *multiple integral* of $f$ over $B$, written\n$$\\int_B f \\, dV = \\int \\cdots \\int_B f(x_1, \\dots, x_n) \\, dx_1 \\cdots dx_n.$$\n\nLet $A \\subseteq \\R^n$ be bounded and let $f \\colon A \\to \\R$.\nLet $f_A$ be the function on a closed box $B \\supseteq A$ that equals $f$ on $A$ and $0$ on $B \\setminus A$.\nThen $f$ is integrable on $A$ if $f_A$ is integrable on $B$, and $\\int_A f \\, dV = \\int_B f_A \\, dV$.\nThis does not depend on the choice of $B$.\n\n## Remarks\n\nFor $n = 1$ this is the [[Riemann integral|0W29TBTY12]]; for $n = 2$ and $n = 3$ it is the [[double integral|0WAR4F23B4]] and the [[triple integral|0698M7C7AX]].\nA bounded set $A$ is [[Jordan measurable|0KTGJW1ZTH]] exactly when the constant $1$ is integrable on $A$, and then $\\int_A 1 \\, dV$ is the Jordan measure of $A$.\n\nExample: $f(x, y) = xy$ on $B = [0, 1]^2$.\nThe function is continuous, so it is integrable, and computing it as an iterated integral gives $\\int_0^1 \\int_0^1 xy \\, dx \\, dy = \\frac{1}{2} \\cdot \\frac{1}{2} = \\frac{1}{4}$.\n\nNon-example: let $f$ on $[0, 1]^2$ be $1$ at points with both coordinates rational and $0$ elsewhere.\nEvery subbox contains points of both kinds, so some tagged partitions of any mesh have Riemann sum $1$ and others have Riemann sum $0$.\nNo number $I$ satisfies the condition, so $f$ is not Riemann integrable.\n",
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