Find the measure of the set of real numbers in $(0,1)$ whose binary expansions contains zeroes in the odd positions, such that $x = 0.k_1k_2k_3\ldots$
Checking odd positions one at a time:
If $k_1 = 0, x \in (0,\frac12)$.
If $k_1,k_2 = 0, x \in (0,\frac{1}{8}) \cup (\frac{2}{8},\frac38)$.
If $k_1,k_2,k_3 = 0, x \in (0,\frac{1}{32}) \cup (\frac{2}{32},\frac{3}{32}) \cup (\frac{8}{32},\frac{9}{32}) \cup (\frac{10}{32}.\frac{11}{32})$.
I'm at a loss for characterizing this in a more concise way, akin to the Cantor set, to the union containing all $x$ such that $k_1,k_2,\ldots,k_n = 0$.