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  "body": "\n## Summary\n\nFor $1 \\le p \u003c \\infty$, the space $L^p(\\mu)$ consists of the measurable functions $f$ on a measure space with $\\int \\abs{f}^p \\, d\\mu \u003c \\infty$.\nTwo functions that agree almost everywhere count as the same element.\nThe norm is $\\norm{f}_p = (\\int \\abs{f}^p \\, d\\mu)^{1/p}$, and with this norm $L^p(\\mu)$ is a Banach space.\nThe case $p = 2$ is a Hilbert space.\nThese spaces are the basic function spaces of analysis, probability and partial differential equations.\n\n## Explanations\n\n### L1 human\n\nThe number $\\norm{f}_p$ measures the size of a function by averaging the $p$-th power of its absolute value.\nA larger $p$ gives more weight to the places where the function is large.\nTwo functions that differ only on a set of measure zero have the same size, so they are treated as the same element.\nFor example, $f(x) = x^{-1/3}$ on $(0, 1]$ lies in $L^2$, since $\\int_0^1 x^{-2/3} \\, dx = 3$, but not in $L^3$, since $\\int_0^1 x^{-1} \\, dx = \\infty$.\n",
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