A Project Report On Hausdorff Measure

Measure Theory is the rigorous mathematical study of the field commonlyknown as Hausdorff measure.Hausdorff measures were introduced as certain lower dimensional measures on ℜwhich allow us to measure “small” subsets in ℜ. The Hausdorff measure and the associated Hausdorff dimension of the set provide a more delicate sense of the size of a set in ℜthan the Lebesguemeasure provides. rnIn this work we study means of constructingHausdorff measures, via the so-called “outer measure” and “Carathéodory measure",which isolate certain small-scale features of complicated sets in a metric space. The construction is explicit andcovered in detail, after which specific instances of constructed measures areinvestigated in depth.

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