Let $U$ be uniformly distributed on the interval $[0,1] $. Consider the digits in the decimal expansion of $U$. Show that the probability of $U$ having a decimal expansion ending with repeating 9’s is zero. Show that this is equivalent to showing that the probability of $U$ having a decimal expansion ending with repeating 0’s is zero.
I got completely lost in this question.May be very new in this measure theoretical course. Please help me out.
HINTS:
Show that the set of numbers in $[0,1]$ with decimal expansions ending in repeating nines is countable. What do you know about the measure of countable sets of real numbers?
Show that if $x\in(0,1)$, then $x$ has a decimal representation ending in repeating nines if and only if it has one ending in repeating zeroes. Example: $0.123999\ldots=0.124000\ldots\;$. In addition, $0$ has a decimal representation ending in repeating zeroes, and $1$ has a decimal representation ending in repeating nines.