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Linear Isometry, Radon-Nikodym Derivative and Isomorphisms

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Let be a measurable space and let be two -finite measures defined on . Suppose and is the Radon-Nikodym derivative of with respect to . Define by Show that is a well-defined linear isometry and is an isomorphism if and only if (i.e are mutually absolutely continuous).
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Linear Isometry, Radon-Nikodym Derivative and Isomorphisms are investigated. The solution is detailed and well presented.

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