Transfinite multifractal dimension spectrums

Williams, Stanley

doi:10.1090/S0002-9947-96-01622-4

Transfinite multifractal dimension spectrums
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by Stanley C. Williams

Trans. Amer. Math. Soc. 348 (1996), 4043-4081

DOI: https://doi.org/10.1090/S0002-9947-96-01622-4

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Abstract:

The first order theory of the decomposition of measures with respect to dimension which has been developed by Kahane, Katznelson, Cutler, and others is extended through transfinite recursion to a $\omega _1$-order theory. Necessary and sufficient conditions for a finite regular Borel measure on $[0,d]^{\omega _1}$ to be a $\omega _1$-order multispectrum for a finite Borel measure on $\mathbb {R}^d$ is given.

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