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  "body": "\n## Statement\n\nLet $X$ be an open subset of $\\C^n$, let $E$ be an [[analytic subset|VMDTGTMN9P]] of $X$, and let $k \\ge 1$ be an integer.\nLet $A$ be an analytic subset of $X \\setminus E$ of [[pure dimension|C2R5Q2Q717]] $k$.\nSuppose that every point of $E$ has a neighbourhood $W$ in $X$ such that $\\mathcal{H}^{2k}(A \\cap W) \u003c \\infty$, where $\\mathcal{H}^{2k}$ is the $2k$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $\\C^n = \\R^{2n}$.\nThen the closure $\\overline{A}$ of $A$ in $X$ is an analytic subset of $X$, of pure dimension $k$.\n\n## Remarks\n\nNo condition on the dimension of $E$ is needed; it is replaced by the condition that $A$ has locally finite $2k$-dimensional volume near $E$.\nThe idea of the proof is to use the volume bound to control generic linear projections to $\\C^k$.\nNear each point of $E$, the closure $\\overline{A}$ then projects properly, with a bounded number of sheets.\nThe symmetric functions of the sheets are bounded holomorphic functions outside a thin set, and they extend.\nExample: let $E = \\set{z_1 = 0}$ in $\\C^2$ and $A = \\set{z_2 = 0,\\ z_1 \\ne 0}$; the area of $A$ in each ball is finite, and indeed $\\overline{A}$ is the $z_1$-axis.\nThe volume hypothesis is needed: for $A = \\set{(z_1, e^{1/z_1}) : z_1 \\ne 0}$ and the same $E$, the closure $A \\cup E$ is not analytic, so by the theorem $A$ has infinite area near every point of $E$.\nThe theorem also holds with $X$ a complex manifold, since the statement is local.\n\n## Explanations\n\n### L1 human\n\nAn analytic set known outside an exceptional analytic set extends across it, as long as it has finite area, or finite volume in the right dimension, near the exceptional set.\nA curve spiralling wildly towards a line, like the graph of $e^{1/z}$ near $z = 0$, has infinite area and does not extend.\nFinite volume rules out this kind of wild behaviour.\n\n### L2 human\n\nBishop's theorem strengthens the Remmert–Stein theorem, in which the exceptional set must have smaller dimension than $A$.\nVolume bounds of this kind come naturally from Wirtinger's inequality and Stokes' theorem, which makes the theorem useful for limits of analytic cycles.\nIt is a main step in proving that limits of complex analytic cycles with bounded volume are analytic.\n",
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