10. Let $0 . \overline{a_{1} a_{2} a_{3} 1}$ be a four-digit decimal pure fraction, where $a_{i}(i=1,2,3)$ can only take 0 or 2. Let $T$ be the number of all such four-digit fractions, and $S$ be the sum of all such four-digit fractions.
10.0.111 1
$a_{1} a_{2} a_{3}$ can be respectively
$$
000,200,020,002,022,202,220,222 \text {. }
$$

a total of 8 kinds. Then $S=0.8888$.