Hausdorff Calculus and Some of Its Applications

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Hausdorff Calculus and Some of Its Applications *)

The concepts of Hausdorff derivative and Hausdorff integral were introduced by S.S. Fu. Further, we find that a Lebesgue singular function can usually be decomposed into the sum of an absolutely continuous function and a singular function in Hausdorff sense. Therefore we obtain a more general theorem of Lebesgue decomposition. And in many cases, a Lebesgue singular function is just an absolutely continuous function in Hausdorff sense, so we have an integral representation. As its application, a better formular of integral expression of the length of an arc is given.