Convolution and representations

Nicolas Bourbaki

Chapter Chapter VIII in Integration II, 2004, pp 95-176 from Springer

Abstract: Abstract Recall (Ch. V, §6, Nos. 1 and 4; Ch. VI, §2, No. 10) that, if X and Y are locally compact spaces, μ a measure on X, and φ a mapping of X into Y, φ is said to be μ-proper if: a) φ is μ-measurable; b) for every compact subset K of Y, −1 φ (K) is essentially μ-integrable. Then the image measure v = φ(μ) on Y exists and has the following property: for a function f on Y, with values in a Banach space or in % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaa % qaaiaadkfaaaaaaa!3891! $$ \overline R $$ , to be essentially integrable for v, it is necessary and sufficient that f ○ φ be so for μ, in which case, % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWdra % qaaiaadAgadaqadaqaaiaadMhaaiaawIcacaGLPaaaaSqaaiaadMfa % aeqaniabgUIiYdGccaWGKbGaamODamaabmaabaGaamyEaaGaayjkai % aawMcaaiabg2da9maapebabaGaamOzamaabmaabaGaeqOXdO2aaeWa % aeaacaWG4baacaGLOaGaayzkaaaacaGLOaGaayzkaaaaleaacaWGyb % aabeqdcqGHRiI8aOGaamizaiabeY7aTnaabmaabaGaamiEaaGaayjk % aiaawMcaaaaa!524B! $$ \int_Y {f\left( y \right)} dv\left( y \right) = \int_X {f\left( {\varphi \left( x \right)} \right)} d\mu \left( x \right) $$

Keywords: Compact Subset; Linear Representation; Compact Group; Compact Space; Haar Measure (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-662-07931-7_2

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