STUDIA MATHEMATICA 204 (1) (2011) Higher order local dimensions and Baire category by Lars Olsen (St. Andrews) Abstract. Let X be a complete metric space and write P (X) for the family of all Borel probability measures on X. The local dimension dim loc (μ; x) of a measure μ ∈P (X) at a point x ∈ X is defined by dim loc (μ; x) = lim rց0 log μ(B(x, r)) log r whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈P (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 of a typical (in the sense of Baire category) measure exists at x. Quite surprisingly, we prove that this is not the case. In fact, we show that the local dimension of a typical measure fails to exist in a very spectacular way. Namely, the behaviour of a typical measure μ ∈P (X) is so extremely irregular that, for a fixed x ∈ X, the local dimension function, r → log μ(B(x, r)) log r , of μ at x remains divergent as r ց 0 even after being “averaged” or “smoothened out” by very general and powerful averaging methods, including, for example, higher order Riesz–Hardy logarithmic averages and Ces`aro averages. 1. Statement of the main result. Recall that genericity in a topolog- ical sense is defined as follows: we say that a typical element x in a complete metric space M has property P if the set of elements in M that do not have property P is meagre (i.e. of the first Baire category). Fix a complete metric space X and write P (X )= {μ | μ is a Borel probability measure on X }, and equip P (X ) with the weak topology. Then P (X ) is a complete metric space. Consequently, we can apply the above definition of topological gener- icity to M = P (X ), allowing us to talk about typical probability measures 2010 Mathematics Subject Classification : Primary 28A80. Key words and phrases : multifractals, local dimension, average local dimension, higher order local dimension, summability method, Riesz–Hardy logarithmic averages, Ces`aro averages, Baire category. DOI: 10.4064/sm204-1-1 [1] c  Instytut Matematyczny PAN, 2011