Geodesic
Some generic properties of concentration dimension of measure
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 211-219.

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Let $K$ be a compact quasi self-similar set in a complete metric space $X$ and let $\mathfrak{M}_{1}(K)$ denote the space of all probability measures on $K$, endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in $\mathfrak{M}_{1}(K)$ the lower concentration dimension is equal to $0$, while the upper concentration dimension is equal to the Hausdorff dimension of $K$.
Sia $K$ un sottoinsieme quasi similare compatto di uno spazio metrico completo. Sia $\mathfrak{M}_{1}(K)$ lo spazio delle misure di probabilità su $K$ munito della metrica di Fortet-Mourier. Si dimostra che per una misura $\mu \in \mathfrak{M}_{1}(K)$ tipica (nel senzo della categoria di Baire) la dimensione inferiore di concentrazione è uguale a zero, invece la dimensione superiore di concentrazione è uguale alla dimensione di Hausdorff dell'insieme $K$. 
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Myjak, Józef; Szarek, Tomasz. Some generic properties of concentration dimension of measure. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 211-219. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_1_a12/

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