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Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $di{m}_{loc}(\mu ;x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $di{m}_{loc}(\mu ;x)=li{m}_{r\searrow 0}\left(log\mu \left(B(x,r)\right)\right)/\left(logr\right)$ whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension...