Lebesgue measure of proper subset $A \subset [0, 1]$. 
Let $A$ be a subset of $[0, 1]$, such that for each $a$ in $A$ $a$ has only finite number of 1 in its expansion with numerial base 3.

Then I tried to calculate measure for its compliment $B$ which consists only of numbers with infintely many units in their expansion. 
I have an idea that measure of $B$ is $1/3$. $B$ cant be covered by intervals of length less then $1/3$.

So I need to prove it formally.
Thanks!
 A: Maybe you know the Cantor set $\mathcal C$ which is known for its property of being an uncountable set of (Lebesgue) measure zero. It also has the property that it consists of all numbers in $[0,1]$ with no $1$ in their ternary expansion (i.e. expressed in base $3$). It is generated by starting with $[0,1]$ iteratively removing the middle third from all remaining intervals.

From this we can build $\mathcal C_k$, the set of number with no $1$ after the first $k$ digits of the ternary expansion, in the following way:
$$\mathcal C_k=\frac1{3^k}\cdot\bigcup_{i=0}^{3^k-1} (\mathcal C+i).$$
All arithmentic operations are considered element-wise. This gives a scaled version of $3^k$ Cantor sets glued together end to end. Why does it work? Note that $\mathcal C+i$ consists of all numbers with no $1$ in the ternary expansion after the seperator (I mean the "$.$" to seperate fractional from integer part, I have no better word), but the $i$ generates all possible combination of the $k$ digits from $\{0,1,2\}$ in front of the seperator. Now, by dividing by $3^k$ we scale the set back to $[0,1]$ and in this way bring all these combinations behind the seperator. Note that $\mathcal C_k$ is a finite union of sets of measure zero, hence itself of measure zero.
Now your set $A$ can be given as
$$A=\bigcup_{k\in\Bbb N} \mathcal C_k.$$
Because finitely many $1$'s means that there must be a last one, e.g. at place $k$, we know that this number is contained in $\mathcal C_k$. So we see that it is contained in $A$. And as a countable union of null sets, it is of zero measure.
