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A metric p which does it is for instance r_2=12-e P(µ1,µ2) f < - f < 1+f -f where Wj, $ = 1, 2.... is a dense set in C0(X, ][gym). Let µ be a Radon measure on X and let p be a real number with 1 < p < oc. , for which f If Ipdµ < oo is denoted by LP(X; µ). As it is well-known LP(X; µ) is a Banach space with norm 1/p (J If I P dµ) II f II LP (X;N,) = II f Il p (3) moreover Cc (X) is dense in LP (X; µ), and the dual space of LP (X; µ) is LP' (X; µ) where p' is the dual exponent of p, P + -, = 1 if p > 1 or p' = oc if p = 1.

X E X and where Mx f f dp = f f f (x, y) dvx XxY X (y) da(x) Y Moreover, if µ(X xY) < oo, we can take. = 7r#Fp where 7r is the linear projection rr(x, y) := x. e. , vx (X) = 1. 1. General Measure Theory 36 2 Weak Convergence Due to its relevance for the sequel, in this section we would like to discuss weak convergence of functions and measures, with the aim of illustrating some of its main features. 1 Weak Convergence of Vector Valued Measures Let X be a locally compact separable metric space. Definition 1.

Observing that RI is the union of a countable family of closed cubes with disjoint interiors and with side one, one easily sees that Lusin's theorem holds for any measurable set A of R', not necessarily of finite measure, if p is the Lebesgue measure C. This remark will be used later. 3 Hausdorff Measures Hausdorff measures are among the most important measures. They allow us to define the dimension of sets in IRn and provide us with s-dimensional measures in JR'1 for any s, 0 < s < n and actually in any metric space.

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3-D Shapes by Marina Cohen

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