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In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of ''s''-dimensional Hausdorff measure. ==Motivation: ''s''-dimensional Hausdorff measure== (詳細はsubset ''E'' of ''X''. Given a number ''s'' ≥ 0, the ''s''-dimensional Hausdorff measure of ''E'', denoted ''μ''''s''(''E''), is defined by : where : ''μ''''δ''''s''(''E'') can be thought of as an approximation to the "true" ''s''-dimensional area/volume of ''E'' given by calculating the minimal ''s''-dimensional area/volume of a covering of ''E'' by sets of diameter at most ''δ''. As a function of increasing ''s'', ''μ''''s''(''E'') is non-increasing. In fact, for all values of ''s'', except possibly one, ''H''''s''(''E'') is either 0 or +∞; this exceptional value is called the Hausdorff dimension of ''E'', here denoted dimH(''E''). Intuitively speaking, ''μ''''s''(''E'') = +∞ for ''s'' < dimH(''E'') for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, ''μ''''s''(''E'') = 0 for ''s'' > dimH(''E'') for the same reason as the 3-dimensional volume of a disc in the Euclidean plane is zero. The idea of a dimension function is to use different functions of diameter than just diam(''C'')''s'' for some ''s'', and to look for the same property of the Hausdorff measure being finite and non-zero. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dimension function」の詳細全文を読む スポンサード リンク
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