Is A_epsilon Non-Empty and Measurable in Lebesgue's Theory?

[fsblajinha]

Is $q_ {1} q_ {2}, ...,$ an enumeration of the rational $[0,1]$. For every $\epsilon> 0$, let $\displaystyle{A _ {\epsilon}: = [0,1]-\bigcup_ {n \geq 1} I_{n}}$, where $I_{n}:= [q_{n} - \frac {\epsilon}{2^{n+1}} , q_{n}\frac{\epsilon}{2^{n + 1}}]\cap [0,1]$.

Show that:

1 - $A _ {\epsilon}$ is measurable, non-empty and empty inside;

2 - $\lambda^{*}(B)=1$, where $\displaystyle B:=\bigcup_{k\geq 1}{A_\frac{1}{k}}$

 

[Opalg]

Is $q_ {1} q_ {2}, ...,$ an enumeration of the rational $[0,1]$. For every $\epsilon> 0$, let $\displaystyle{A _ {\epsilon}: = [0,1]-\bigcup_ {n \geq 1} I_{n}}$, where $I_{n}:= [q_{n} - \frac {\epsilon}{2^{n+1}} , q_{n}\frac{\epsilon}{2^{n + 1}}]\cap [0,1]$.

Show that:

1 - $A _ {\epsilon}$ is measurable, non-empty and empty inside;

2 - $\lambda^{*}(B)=1$, where $\displaystyle B:=\bigcup_{k\geq 1}{A_\frac{1}{k}}$

Hi fsblajinha, and welcome to MHB!

Can you tell us what progress you have made with this problem, and where you need assistance?

 

[fsblajinha]

Hi fsblajinha, and welcome to MHB!

Can you tell us what progress you have made with this problem, and where you need assistance?

I can not prove that it has empty interior! Thank you!

 

[Opalg]

I can not prove that it has empty interior!

Hint: the complement of $A _ {\epsilon}$ contains every rational point in the interval.