For all $F\subset \mathbb R^n$, we define $$\mathcal H_\delta^s(F) = \inf\left\{\sum_{i=1}^\infty |U_i|^s : F \subset \bigcup_{i=1}^\infty U_i, 0 \le |U_i| \le \delta \right\}$$ where $|U_i|$ is the diameter of the set $U_i$, defined in the usual way. We also define $$\mathcal H^s(F) = \lim_{\delta\to 0} \mathcal H_\delta^s(F)$$ for every $F\subset\mathbb R^n$.
To show that $\mathcal H^s(F)$ is a measure, it is enough to prove that the following three properties are satisfied:
(a) $\mathcal H^s(\varnothing) = 0$,
(b) $A \subset B \implies \mathcal H^s(A) \le \mathcal H^s(B)$,
(c) $\mathcal H^s\left(\bigcup_{i=1}^\infty A_i\right) \le \sum_{i=1}^\infty \mathcal H^s(A_i)$ with equality if $A_i$'s are disjoint Borel sets.
I need help with (c).
(c) Following the ideas of the previous two parts, I would first like to show that $\mathcal H^s_\delta\left(\bigcup_{i=1}^\infty A_i\right) \le \sum_{i=1}^\infty \mathcal H^s_\delta(A_i)$ and then take limits as $\delta\to 0$. Consider $A_i$ for some $i$. Let $\{U_{ij}\}_{j=1}^\infty$ be a $\delta$-cover of $A_i$, i.e. $A_i \subset \bigcup_{j=1}^\infty U_{ij}$. Then, $$\bigcup_{i=1}^\infty A_i \subset \bigcup_{i=1}^\infty \bigcup_{j=1}^\infty U_{ij}$$ i.e. the union of all covers is a cover for the union of $A_i$. How do I proceed from here? I also need to show that equality holds if $A_i$'s are disjoint Borel sets.
Thanks for your help!