On the Divergence of General Local and Fractal Dimensions of Typical Measures

Consider \(X\) as a complete separable metric space. This paper demonstrates that, for a typical probability measure, the general lower local dimension is consistently zero, while the general upper local dimension remains infinite. Additionally, the general lower local dimension is almost always zero, and given certain additional conditions on \(X\), a corresponding outcome is established for the general upper local dimension. Furthermore, we establish that the overall Hausdorff dimension of a typical measure on the space \(X\) is consistently zero within any compact space. Simultaneously, under specific additional conditions, we show that the general packing dimension surpasses a predefined value \(s > 0\). More precisely, we show that the local dimension of a typical measure fails to exist. Specifically, the behavior of a typical measure \(ν\) is so extremely irregular that, for a fixed point \(x ∈ X\), the generalized local dimension function \[ r ↦ φ ( ν(B_r(x)) )/ψ(r), \] of \(ν\) at \(x\), with respect to the functions \(φ\) and \(ψ\), remains divergent as \(r → 0\), even after being ''averaged'' or ''smoothed out'' by general and powerful averaging methods. These include, for example, higher-order Riesz-Hardy logarithmic averages and Cesàro averages.