Help me make sense of t-Hausdorff measures

Question

I am so confused about the definition of Hausdorff measures. In the book I'm following, it is defined as such:
Given our set F, we define the quantity $$H_{\delta}^t=\inf\{\sum_{i=1}^{\infty}|U_i|^t:\{U_i\} \textit{ is a } \delta \textit{-cover of F} \}$$ where a delta cover is an infinite collection of sets with diameter smaller than delta such that F is contained it their reunion, and || denotes the diameter. Then the author defines the Hausdorff measure as the limit of this quantity as delta goes to infinity, and claims this limit is infinity unless $t$ is above some critical value which he then proceeds to call the Hausdorff dimension of the set.
This is where I'm lost. How could the infimum of that set ever be zero? When I increase delta, shouldn't the diameters of the members of the cover also blow up? Can someone clarify?