Is it possible to define a number, order preserving map, or functional $\alpha$ such that $\alpha\mathbb{N}$ has Lebesgue measure $1$?

Question

Is it possible to define a number, order preserving map, or functional $\alpha$ such that $\alpha\mathbb{N}=\{x|x=\alpha n $ for $n\in\mathbb{N}\}$, and the Lebesgue outter measure $m^*(\alpha\mathbb{N})=1$?

What if there is a further restriction such that $\alpha\mathbb{N}\in[0,1)$?

(I was thinking about some number like $\frac{1}{\omega}$)

Answer 1

$\mathbb N$ is countable, and any countable set has Lebesgue (outer) measure $0$. Therefore in order for this to work, you would have to pick a one-to-many mapping, and specifically one that maps a single point to uncountably many points. I’m not 100% sure what you’re after, but I doubt that that’s what you’re after. So, I’m going to say the answer is no you cannot.