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boolemap - The Boole Transformation Jonathan L.F King...

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The Boole Transformation Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 20 November, 2006 (at 18:39 ) Abstract: The Boole map preserve Lebesgue measure on the one-point compactification ˙ R of R . The scaled Boole map preserves the finite measure having density 1 / [ x 2 + 1]. A condition for an endomorphism of R to be measure-preserving. Fix an open subset X R . When does a continuously differentiable ( not necessar- ily invertible ) map f : X preserve Lebesgue measure? ( N.B. For simplicity, we assume henceforth that f is every- where finite-to-one. ) Evidently f is measure-preserving exactly when x X : X r f 1 ( x ) 1 | f 0 ( r ) | = 1 . For consider a small interval I centered at x . The inverse images of x , write them r 1 < . . . < r N , have some minimum distance between them and so, if I is small enough, then f 1 ( I ) consists of N disjoint intervals whose lengths are essentially Len( I ) /f 0 ( r n ) N n =1 . The intervals have “exactly” these lengths when I is infinitesimal.
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