A question about finding Lebesgue measure of a specific set I am unable to find

Question

This is a quiz question of previous year asked in my measure theory exam and I am unable to solve it.

Let $k$ be a positive integer and let $$S_{k} = \{x \in [0, 1] | \text{ a decimal expansion of $x$ has a prime digit at its $k$-th place}\}.$$ Then the Lebesgue measure of $S_{k} $ is?

I know the definition of Lebesgue measure and I self studied it from Tom M Apostol Mathematical analysis and Walter Rudin (very few online classes).

But unfortunately, I don't know how this particular question can be approached and I am struck.

Any help would be really appreciated.

Answer 1

Let $\mu$ be the Lebesgue measure on $[0,1]$. Let $I’=\{x\in (0,1]: x\mbox{ has a unique decimal expansion}\}$. Since $[0,1]\setminus I’$ is a set of numbers admitting a finite decimal expansion, it is countable, so $\mu(I’)=\mu([0,1])=1$.

For each digit $i$, let $I’_i=\{x\in I’: \mbox{ a decimal expansion of $x$ has a digit $i$ at its $k$-th place}\}$. It is easy to see that each set $I’_i$ is Lebesgue measurable and $I’$ is a disjoint union of $ I’_i $. For each $i$ we have $I'_i=I’_0+i\cdot 10^{-k}$. Since $\mu$ is translation-invariant, (that is $\mu(A+x)=\mu(A)$ for each measurable subset $A$ of $\Bbb R$ and $x\in\Bbb R$), we have $\mu(I’_i)= \mu(I’_0)$ for each $i$. Since $\sum_{i} \mu(I’_i)= \mu(I’)=\mu([0,1])=1$, we obtain that $\mu(I’_i)=1/10$ for each $i$. Then $\mu(S_k)=\sum_{i\mbox{ is prime}}\mu(I’_i)=\mu(I’_2)+ \mu(I’_3)+\mu(I’_5)+ \mu(I’_7)=4/10$.

Answer 2

Another way of asking the same question would be: What is the probability that $x \in [0,1]$ has a prime as its $k$-th digit? Can you now solve it?

(There is some issue with numbers that have more than one decimal expansion, but I assume this subtlety is not intended in the quiz.)

Answer 3

Often I've found that when confronted with a set whose measure seems impossible to guess, it's measure zero. Even if I have no intuition as to why a set "aught" to be measure zero; if I'm super stumped, it's a good place to start investigating.

Can you put an arbitrarily small open set around every element of $S_k$? Or at least, can you cover $S_k$ by $\varepsilon$-balls?

EDIT: The above advice is still good advice in general, but we can actually just brute force this. Note that $S_k$ only has primes at the $k$th place, not any place like I was originally thinking.

The only prime digits are 2, 3, 5, and 7. So $S_1$ is $[0.2, 0.3)\cup[0.3,0.4)\cup[0.5,0.6)\cup[0.7,0.8)$ and $\mu(S_1)=4\times10^{-1}$. Follow the pattern to finish the problem.