Fundamental theorems of calculus for packing measures on the real line. (English) Zbl 0747.28002
Summary: The fundamental theorems of calculus are extended to the treatment of packing measures on the real line. These are related to the corresponding result for Hausdorff measures. We prove that the centered inner-envelope derivative and the outer-envelope derivative of a continuous increasing function on an interval differ, but in a certain sense they are almost everywhere linear and the theorems are true for \(h\)-continuous almost \(h\)-singular functions of bounded variation.
MSC:
| 28A78 | Hausdorff and packing measures |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
| 26A42 | Integrals of Riemann, Stieltjes and Lebesgue type |
Keywords:
fundamental theorems of calculus; packing measures on the real line; Hausdorff measures; centered inner-envelope derivative; outer-envelope derivative; \(h\)-continuous almost \(h\)-singular functions of bounded variationReferences:
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