Lebesgue integral over a subset of the measure space

Let $(X,\\mathfrak{B},\\mu)$ be a measure space and $A\\in\\mathfrak{B}$ .

Let $s\\colon X\\to[0,\\infty]$ be a simple function. Then $\\displaystyle\\int_{A}s\\,d\\mu$ is defined as $\\displaystyle\\int_{A}s\\,d\\mu:=\\int_{X}\\chi_{A}s\\,d\\mu$ , where $\\chi_{A}$ denotes the characteristic function of $A$ .

Let $f\\colon X\\to[0,\\infty]$ be a measurable function and
$S=\\{s\\colon X\\to[0,\\infty]~{}~{}|~{}~{}s\\text{ is a simple function and }s\\leqf\\}$ . Then $\\displaystyle\\int_{A}f\\,d\\mu$ is defined as $\\displaystyle\\int_{A}f\\,d\\mu:=\\sup_{s\\in S}\\int_{A}s\\,d\\mu$ .

By the properties of the Lebesgue integral of nonnegative measurable functions (property 3), we have that $\\displaystyle\\int_{A}f\\,d\\mu=\\int_{X}\\chi_{A}f\\,d\\mu$ .

Let $f\\colon X\\to[-\\infty,\\infty]$ be a measurable function such that not both of $\\displaystyle\\int_{A}f^{+}\\,d\\mu$ and $\\displaystyle\\int_{A}f^{-}\\,d\\mu$ are infinite. (Note that $f^{+}$ and $f^{-}$ are defined in the entry Lebesgue integral.) Then $\\displaystyle\\int_{A}f\\,d\\mu$ is defined as $\\displaystyle\\int_{A}f\\,d\\mu:=\\int_{A}f^{+}\\,d\\mu-\\int_{A}f^{-}\\,d\\mu$ .

By the properties of the Lebesgue integral of Lebesgue integrable functions (property 3), we have that $\\displaystyle\\int_{A}f\\,d\\mu=\\int_{X}\\chi_{A}f\\,d\\mu$ .

单词	LebesgueIntegralOverASubsetOfTheMeasureSpace
释义	Lebesgue integral over a subset of the measure space Let $(X,\\mathfrak{B},\\mu)$ be a measure space and $A\\in\\mathfrak{B}$ . Let $s\\colon X\\to[0,\\infty]$ be a simple function. Then $\\displaystyle\\int_{A}s\\,d\\mu$ is defined as $\\displaystyle\\int_{A}s\\,d\\mu:=\\int_{X}\\chi_{A}s\\,d\\mu$ , where $\\chi_{A}$ denotes the characteristic function of $A$ . Let $f\\colon X\\to[0,\\infty]$ be a measurable function and $S=\\{s\\colon X\\to[0,\\infty]~{}~{}\|~{}~{}s\\text{ is a simple function and }s\\leqf\\}$ . Then $\\displaystyle\\int_{A}f\\,d\\mu$ is defined as $\\displaystyle\\int_{A}f\\,d\\mu:=\\sup_{s\\in S}\\int_{A}s\\,d\\mu$ . By the properties of the Lebesgue integral of nonnegative measurable functions (property 3), we have that $\\displaystyle\\int_{A}f\\,d\\mu=\\int_{X}\\chi_{A}f\\,d\\mu$ . Let $f\\colon X\\to[-\\infty,\\infty]$ be a measurable function such that not both of $\\displaystyle\\int_{A}f^{+}\\,d\\mu$ and $\\displaystyle\\int_{A}f^{-}\\,d\\mu$ are infinite. (Note that $f^{+}$ and $f^{-}$ are defined in the entry Lebesgue integral.) Then $\\displaystyle\\int_{A}f\\,d\\mu$ is defined as $\\displaystyle\\int_{A}f\\,d\\mu:=\\int_{A}f^{+}\\,d\\mu-\\int_{A}f^{-}\\,d\\mu$ . By the properties of the Lebesgue integral of Lebesgue integrable functions (property 3), we have that $\\displaystyle\\int_{A}f\\,d\\mu=\\int_{X}\\chi_{A}f\\,d\\mu$ .
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