Lebesgue measure of a set $A \subset [0;1]$. A set consists of numbers (in interval $[0;1]$) which, when written in a decimal form, have digit $2$ occurring "before" digit $3$ can be found. It sounds awkwardly, but I hope you get what I mean:
$0.1213 \in A \\ 
0.1231 \in A \\
0.1211 \in A \\
0.1111 \notin A \\
0.1321 \notin A \\
0.1312 \notin A$
I saw a few somewhat similar examples, so I suspect that the set is indeed Lebesgue measurable, but I'm struggling to find a union of (hopefully distinct) intervals for it...
 A: Longish Hint: It is probably easier to compute the Lebesgue measure of the complement $[0,1] \setminus A$ of this set.  A real number in $[0,1]$ will be in the complement iff it has a decimal exapansion in which $3$ occurs before any occurrence of $2$[*].  So here are pairwise disjoint intervals which make up the set:


*

*$[0.3,0.4]$;

*$[0.03,0.04]$, $[0.13,0.14]$, $[0.43,0.44]$, $[0.53,0.54]$, $[0.63,0.64]$, $[0.73,0.74]$, $[0.83,0.84]$, $[0.93,0.94]$,  (the interval $[0.33,0.34]$ is a subset of the interval from the previous level, so we omit it here);

*$[0.ij3,0.ij4]$ where $i,j \in \{ 0 , 1 , 4, 5, 6, 7, 8, 9 \}$.

*$\vdots$


In general, for the $n$th level there will be $8^{n-1}$ intervals each of length $10^{-n}$, given a total measure of $8^{n-1} 10^{-n}$.  From here it should be easy to compute the Lebesgue measure of $[0,1] \setminus A$, and thus also the Lebesgue measure of $A$.

[*] There might be a slight quibble about this, depending on whether you see $0.4000\ldots$ as belonging to $A$, or $0.3999\ldots$ as belonging to $[0,1] \setminus A$.  However, as only rational numbers can have two different decimal expansions the Lebesgue measure will be the same with either choice.  (I'm opting for $0.4000\ldots = 0.3999\ldots \notin A$.)


Addendum: Thanks to comments from the OP, I have realised that the intended definition of the set $A$ is as follows: $x \in [0,1]$ belongs to $A$ iff if has a decimal expansion in which $2$ occurs, and there are no occurrences of $3$ before the first occurrence of a $2$.  However the analysis done above will work with only minor modifications to determine the measure of the set $A$ (and this we will compute directly).  A collection of pairwise disjoint intervals making up $A$ are:


*

*$[0.2,0.3]$;

*$[0.02,0.03]$, $[0.12,0.13]$, $[0.42,0.43]$, $[0.52,0.53]$, $[0.62,0.63]$, $[0.72,0.73]$, $[0.82,0.83]$, $[0.92,0.93]$,  (the interval $[0.22,0.23]$ is a subset of the interval from the previous level, so we omit it here);

*$[0.ij2,0.ij3]$ where $i,j \in \{ 0 , 1 , 4, 5, 6, 7, 8, 9 \}$.

*$\vdots$


The connection between this and the above should be clear.
A: So, the numbers 00-99 are 100 combinations,
and there are 19 numbers that contains 3 somewhere. (3 first, 10 combinations, 3 last, 10 combinations, but one combination is counted twice).
Therefore, 19/100 is the measure of your set.
Finding the intervals should be straightforward.
