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In [[size theory]], the '''size function''' <math>\ell_{(M,\varphi)}:\Delta^+=\{(x,y)\in \mathbb{R}^2:x<y\}\to \mathbb{N}</math> associated with the [[size pair]] <math>(M,\varphi:M\to \mathbb{R})</math> is defined in the following way. For every <math>(x,y)\in \Delta^+</math>, <math>\ell_{(M,\varphi)}(x,y)</math> is equal to the number of connected components of the set
<math>\{p\in M:\varphi(p)\le y\}</math> that contain at least one point at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>x\ </math>
<ref name="dAFrLa06">Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.</ref>. The concept of size function can be easily extended to the case of a measuring function <math>\varphi:M\to \mathbb{R}^k</math>, where <math>\mathbb{R}^k</math> is endowed with the usual partial order.
<ref name="BiCeFr08">Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi, ''Multidimensional size functions for shape
comparison'', Journal of Mathematical Imaging and Vision 32:161-179, 2008.</ref>
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