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  "body": "\n## Definition\n\nLet $n, m \\ge 1$, and let $\\Omega \\subseteq \\R^n$ be a Lebesgue measurable set.\nLet $C_0(\\R^m)$ be the space of continuous functions $\\varphi \\colon \\R^m \\to \\R$ that tend to $0$ at infinity.\nA *Young measure* on $\\Omega$ with values in $\\R^m$ is a family $(\\nu_x)_{x \\in \\Omega}$ of Borel [[probability measures|059TJN0NZ1]] on $\\R^m$.\nIt must satisfy one condition: for every $\\varphi \\in C_0(\\R^m)$ the function\n$$x \\mapsto \\int_{\\R^m} \\varphi \\, d\\nu_x$$\nis Lebesgue [[measurable|YYNCJSR0PJ]] on $\\Omega$.\n\n## Remarks\n\nA sequence of measurable $u_k \\colon \\Omega \\to \\R^m$ *generates* $(\\nu_x)$ if $\\varphi(u_k) \\to \\int \\varphi \\, d\\nu_x$ weakly-$*$ in $L^\\infty(\\Omega)$ for every $\\varphi \\in C_0(\\R^m)$.\nThe fundamental theorem on Young measures says that every sequence bounded in $L^\\infty(\\Omega; \\R^m)$ has a subsequence that generates a Young measure.\nExample: a measurable $u \\colon \\Omega \\to \\R^m$ gives the Young measure $\\nu_x = \\delta_{u(x)}$, since $\\int \\varphi \\, d\\nu_x = \\varphi(u(x))$ is measurable.\nExample: $u_k(x) = \\sgn(\\sin(2 \\pi k x))$ on $(0,1)$ generates the constant family $\\nu_x = \\frac12 (\\delta_{-1} + \\delta_1)$.\nNon-example: if $g \\colon \\Omega \\to \\R^m$ is not measurable, then $\\nu_x = \\delta_{g(x)}$ is not a Young measure.\nSome open $U$ has a non-measurable preimage, and $1_U$ is an increasing limit of functions $\\varphi_j \\in C_0(\\R^m)$, so some $\\varphi_j \\circ g$ is not measurable.\n",
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