##  [Hausdorff Measure](/hausdorff-measure) 

  ##  [Hausdorff Measure](https://natural.quantumdictionary.io/hausdorff-measure-0) 

  

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**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A family of outer measures on a metric space parameterized by a nonnegative real s, constructed by covering sets with countably many sets of small diameter and taking the infimum of the sum of diameters^s (Carathéodory construction); for each s this yields the s-dimensional Hausdorff measure, which generalizes length, area, and volume and identifies fractal dimensions.

 

 

 

 

 





 

 



 ##  [Hausdorff Measure](https://mathlogic.quantumdictionary.io/hausdorff-measure-1) 

  

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- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

For each nonnegative real s, the Hausdorff s-measure is an outer measure defined on a metric space by taking the limit as δ→0 of the infimum of sums ∑ diam(U_i)^s over countable covers {U_i} with diam(U_i)≤δ; it generalizes length, area, and volume and detects sizes at non-integer (fractal) dimensions.

 

 

 

 

 





 

 



 ##  [Hausdorff Measure](https://puremath.quantumdictionary.io/hausdorff-measure-2) 

  

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- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A family of outer measures on a metric space defined for each dimension parameter d≥0 by covering sets with arbitrarily small diameter and summing the dth power of those diameters (possibly with a constant); these outer measures generalize length, area and volume and detect fractal scaling.