For $E$ a (totally) bounded subset of $X$ and $s>0$, the Hewitt-Stromberg pre-measure is defined as follows, $$ \overline{\mathscr{V}}^s(E)=\limsup_{\delta\to0} M_\delta(E) \;(2\delta)^s, $$ where the packing number $M_\delta(E)$ of $E$ is given by \begin{equation*} \begin{split} M_\delta(E) = & \sup\Biggl\{\sharp\{I\}\; \Bigl|\; \Big(B(x_i, \delta) \Big)_{i\in I} \; \text{is a family of closed balls} \\ &\qquad\qquad\quad\text{with}\; x_i \in E \; \text{and}\; B(x_i, \delta)\cap B(x_j, \delta)=\emptyset\; \text{for}\; i\neq j\Biggl\}. \end{split} \end{equation*}
Now, we define the $s$-dimensional Hewitt-Stromberg measure as follows $$ {\mathscr{V}}^s(E)=\inf\left\{\sum_i \overline{\mathscr{V}}^s(E_i)\;\Big|\;\;E\subseteq\bigcup_i E_i\right\}. $$
We will construct two positively separated sets $E$ and $\widetilde{E}$, and show that there exists $s>0$ such that $${\mathscr{V}}^s( E \cup \widetilde{E})\neq {\mathscr{V}}^s( E ) + {\mathscr{V}}^s( \widetilde{E} ).$$
First, we consider the following two IFS's on $[0,1]$, namely $$S=(S_1,S_2)\quad\text{ where}\;\; S_1(x)=\frac x3 \;\;\text{and}\;\; S_2(x)=\frac x3+\frac 23,$$ and $$T=(T_1,T_2)\quad\text{where}\;\; T_1(x)=\frac x6\;\;\text{and}\;\; T_2(x)=\frac x6+\frac 56.$$
Next, construct the sets $E$ and $F$ as follows. Fix a sequence $n_1,n_2,n_3,...$ of positive integers. We construct the set $E$ as follows. Let \begin{eqnarray*} E_1 &=& { S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots \circ S_{i_{1,n_1}}([0,1]) },\\ E_2 &=& { S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots \circ S_{i_{1,n_1}}\circ T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots\circ T_{j_{1,n_1}}([0,1])},\\ E_3 &=& { S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots \circ S_{i_{1,n_1}} \circ T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots\circ T_{j_{1,n_1}}\circ S_{i_{2,1}}\circ S_{i_{2,2}}\circ\ldots\circ S_{i_{2,n_2}}([0,1]) },\\ E_4 &=& S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots \circ S_{i_{1,n_1}} \circ T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots\circ T_{j_{1,n_1}}\circ S_{i_{2,1}}\circ S_{i_{2,2}}\circ\ldots\circ S_{i_{2,n_2}}\circ\\&&\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad T_{j_{2,1}}\circ T_{j_{2,2}}\circ\ldots\circ T_{j_{2,n_2}}([0,1]),\\ &\vdots& \end{eqnarray*} where the indices $i_{1,1},...,i_{1,n_1}...$ takes on all values of $1,2.$ Continue like this, and put $$E = \bigcap_n \bigcup_{I\in E_n} I.$$
Next, construct the set $F$ as follows. Let \begin{eqnarray*} F_1 &=& { T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots \circ T_{j_{1,n_1}}([0,1])},\\ F_2& =& { T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots \circ T_{j_{1,n_1}}\circ S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots\circ S_{i_{1,n_1}}([0,1]) },\\ F_3& = &{ T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots\circ T_{j_{1,n_1}}\circ S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots\circ S_{i_{1,n_1}}\circ T_{j_{2,1}}\circ T_{j_{2,2}}\circ\ldots\circ T_{j_{2,n_2}}([0,1]) },\\ F_4 &=&T_{j_{1,1}}\circ T_{j_{1,2}}\circ\ldots\circ T_{j_{1,n_1}}\circ S_{i_{1,1}}\circ S_{i_{1,2}}\circ\ldots\circ S_{i_{1,n_1}}\circ T_{j_{2,1}}\circ T_{j_{2,2}}\circ\ldots\circ T_{j_{2,n_2}}\circ\\&&\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\qquad\quad\quad\qquad\qquad\quad\qquad S_{i_{2,1}}\circ S_{i_{2,2}} \circ\ldots\circ S_{i_{2,n_2}}([0,1]),\\ &\vdots& \end{eqnarray*} where the indices $i_{1,1},...,i_{1,n_1}...$ takes on all values of $1,2.$ Continue like this, and put $$ F = \bigcap_n \bigcup_{J\in F_n} J.$$
Now, it is clear that the sets $E$ and $\widetilde{E}:=F+2$ are positively separated, and if we choose the integers $n_1,n_2,n_3,...$ are sufficiently large, then $${\mathscr{V}}^s( E \cup \widetilde{E})< 2 \leq {\mathscr{V}}^s( E ) + {\mathscr{V}}^s( \widetilde{E} )\quad\text{for} \quad s=\frac{2\log2}{\log 3+\log 6}.$$
I need the proof of the last claimed inequalities????