Lebesgue Measure of $S_k$

Question

let $S_k$ is the set of elements of $(0,1)$ whose kth position is prime. Then what's is the lebesgue measure of $S_k$

I don't know how to approach the question... Any help leading to answer is deeply appreciable Thanks

Can you provide the context of the problem? Where did you find this problem? — Shashi, Dec 17 '17 at 17:22
The $k$th position of $a$ is prime $\iff$ the $k$th element of the decimal expansion of $a$ is prime? — eepperly16, Dec 17 '17 at 17:35

mephistolotl · Accepted Answer · 2017-12-18T02:40:51.717

0

$S_1$ is the set of numbers in $(0,1)$ with prime first digit. The only prime digits are 2, 3, 5, and 7, so this is all numbers that start with $0.2$. $0.3$, $0.5$. or $0.7$. That means

$$ S_1 = [0.2,0.3) \cup [0.3,0.4) \cup [0.5,0.6) \cup [0.7,0.8) , $$

and $\mu(S_1) = \frac{4}{10}$.

What is $S_2$?

There is technically an ambiguity: $0.3999\dots = 0.4$, so is it in $S_1$ or not? However, this doesn't affect the answer, why?

edited Dec 18 '17 at 02:40

answered Dec 17 '17 at 18:25

mephistolotl

663

$\mu(S_k) = \frac{4}{10}$ why ? It is the measure of $s_1$ right? and $\mu(S_k) = ({\frac{4}{10}})^k$ because we can further divide those intervals as k increases – Dec 18 '17 at 02:26
That was a bad typo on my part. I meant $\mu(S_1)$. – mephistolotl Dec 18 '17 at 02:41
First convince yourself what $S_2$ and $S_3$ look like. – mephistolotl Dec 18 '17 at 02:42

Lebesgue Measure of $S_k$

1 Answers1