I was a little unsure on this problem. I do have some ideas though. The way I thought of translation invariant is that you can take an interval and shift it, and in the process is will still be the same length. But in this case shifting may change the length since elements that do not contain a negative power are zero. I would appreciate the help!
The measure $\delta_x$ is the measure given by $$\displaystyle \delta_x(E) = \left\{\begin{array}{ll} 1 & x \in E \\ 0 & x \notin E \end{array} \right.$$ Your set function $d$ satisfies $$\mu(E) = \sum_{k =1}^\infty 2^{-k} \delta_{2^{-k}}(E)$$ Can you show a countable sum of measures is a measure? Also note that $$d((0,1]) = \sum_{k=1}^\infty 2^{-k} = 1.$$ Non-invariance under translations is easy to see: $d(\{1\}) \not= d(\{1\} - \frac 12)$, for instance.