User:IntegralPython/sandbox/Fractal measure: Difference between revisions

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{{further|Packing dimension}}

Just as the packing dimension is in some ways a dual to the Hausdorff dimension, the packing measure is a counterpart to the Hausdorff measure. The packing measure is defined informally as the measure of "packing" a set with open balls, and calculating the measure of those balls. In contrast to Hausdorff measure, which covers the set being measured and can have intersecting covers, the packing measure requires no balls to intersect and becomes smaller than the Hausdorff measure.

Let (''X'', ''d'') be a metric space with a subset ''S'' ⊆ ''X'' and let ''s'' ≥ 0. We take a "pre-measure" of ''S'', defined to be