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The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is  for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as

The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is  for <math>d \geq 0 </math>, we define the d-dimensional Hausdorff outer measure of S as

豪斯多夫外测度不同于无界的豪斯多夫，因为我们不考虑 s 的所有可能，我们看到当球的大小为零时会发生什么。这是为了数学  <math>d \geq 0 </math>，我们定义了  ''S''的 ''d''维豪斯多夫Hausdorff 外测度为

豪斯多夫外测度不同于无界的豪斯多夫，因为我们不考虑 s 的所有可能，我们看到当球的大小变为零时会发生什么。对于 <math>d \geq 0 </math>，我们定义了  ''S''的 ''d''维豪斯多夫Hausdorff 外测度为

:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>

:<math> \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 < r_i < r\Bigr\}.</math>