While reading Fractal Geometry by K.Falconer, I have came across a few applications of the Egoroffs Theorem in some proofs. But in none of them, the theorem is used in any easy format. Below I mention one such:
In the fifth chapter,- 'Local structure of Fractals' there is a theorem which states that, a s-set $F$ in $\mathbb{R}^2$ is irregular if, $0<s<1$. So in the proof it is claimed that the density doesn't exist almost everywhere in $F$, and to prove this , forcefully it is assumed that the statement is not true, that is there is a set $F_1 \subset F$, with positive measure where the density exists, i.e $D^s (F,x)$ exist in $F_1$, and hence by a previous proposition $$D^s (F,x) \geq 2^{(-s)}>2^{-1}$$ as $0<s<1$.
And then the $\bf{Egoroff's Theorem}$ is used to find $r_0>0$ and a Borel set $E \subset F_1 \subset F$ with $\mathcal{H}^s (E)>0$ such that,
$\forall x \in E$ and $r<r_0$
$$\mathcal{H}^s(F \cap B(x,r) ) > \frac{(2r)^s}{2}$$.
But this application of Egoroff's Theorem is in no way even nearly close to my understanding of the theorem, because in my understanding the theorem tells that if a sequence of functions $f_k$ converges point-wise to a function $f$ with domain, a Borel set $D$ of finite measure, then for any $\epsilon >0 $ there is a Borel subset $E$ of $D$ such that measure of $(D \setminus E)$ is less than $\epsilon$ and the convergence is uniform in $E$.
I cannot figure out how this version of the theorem is applied on the above proof. I am thinking if anyone can say something on this. Thank you.