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Fractal measure is a generalization of the concepts of length, area, and volume to non-integer dimensions, especially in application towards fractals. There is no unique fractal measure, in part although not entirely due to the lack of a unique definition of fractal dimension; the most common fractal measures include the hausdorff measure and the packing measure, based off of the hausdorff dimension and packing dimension respectively.[1] Fractal measures are measures in the sense of measure theory, and are usually defined to agree with the n-dimensional Lebesgue measure when n is an integer.[2] Fractal measures find application in the study of fractal geometry, as well as in physics and biology through the study of fractal derivatives.[3] Fractal measure can be used to define the fractal dimension or vice versa.

Hausdorff measure

Packing measure

References

  1. ^ http://www.diva-portal.org/smash/get/diva2:22333/FULLTEXT01
  2. ^ https://link.springer.com/chapter/10.1007/978-1-4757-2958-0_1
  3. ^ Chen, W. (2006). "Time–space fabric underlying anomalous diffusion". Chaos, Solitons and Fractals. 28 (4): 923–929. arXiv:math-ph/0505023. Bibcode:2006CSF....28..923C. doi:10.1016/j.chaos.2005.08.199. S2CID 18369880.

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