Lemma showing we can approximate the Hausdorff s-dimensional measure doesn't make any sense

Question

I have been reading (and re-reading, and re-re-reading) Lemma 1.7 from The Geometry of Fractal Sets by K. J. Falconer, and I don't understand what it says. It is prefaced with the following remark:

The next lemma states that any attempt to estimate the Hausdorff measure of a set using a cover of sufficiently small sets gives an answer not much smaller than the actual Hausdorff measure.

The statement of the lemma is as follows.

Let $E$ be $\mathscr{H}^s$-measurable with $\mathscr{H}^s(E)<\infty$, and let $\epsilon > 0$. Then there exists $\rho>0$ (dependent only on $E$ and $\epsilon$), such that for any collection of Borel sets $\{U_i\}_{i=1}^\infty$ with $0<|U_i|\leq \rho$ we have $$\mathscr{H}^s(E\cap (\cup_i U_i))<\sum_i|U_i|^s+\epsilon.$$

I really don't understand what's going on here. A few questions come to mind:

1. Don't we already know by definition of $\mathscr{H}^s$ that it is the limiting case of the estimate of the s-measure of a set by small enough sets?
2. By monotonicity, we have $\mathscr{H}^s(\cup_i U_i)>\mathscr{H}^s(E\cap (\cup_i U_i))$, and $\mathscr{H}^s$ is just the limit as $\delta \to 0$ of the infimum of all covers of a set with each element of the cover having a diameter of no more than $\delta$. So, shouldn't it be immediate that $\sum_i|U_i|>\mathscr{H}^s(\cup_i U_i)>\mathscr{H}^s(E\cap (\cup_i U_i))$? If not, what am I missing?
3. Why do the sets need to be Borel? Is this a necessary condition?

I don't understand the proof at all, but I'm hoping that it will be easier once I can really figure out what the lemma is saying. Any and all help will be appreciated.

Answer 1

The issue is exactly that the definition of $\mathscr{H}^s$ involves a limit. You wrote: "So, shouldn't it be immediate that $\sum_i|U_i|>\mathscr{H}^s(\cup_i U_i)\ldots$?"

The answer is negative, because $\mathscr{H}^s$ is a limit as $\delta \to 0$ where the covering sets are required to have diameter less thatn $\delta$. As you decrease $\delta$, fewer sets are allowed in the cover so you get a higher value for the infimum over covers.

The book you are reading is excellent, but emphasizes the technical points, so perhaps it is not the easiest choice to get into the subject. Falconer's later book [1] is a bit more user-friendly, you can read a few chapters there (or in [2]), and then get back to "The Geometry of Fractal Sets".

[1] Falconer, Kenneth. Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 2004.

[2] Bishop, C.J. and Peres, Y., 2017. Fractals in probability and analysis (Vol. 162). Cambridge University Press. https://www.yuval-peres-books.com/fractals-in-probability-and-analysis/