Question

find a measer dense subset in r2

Answer 1

Answer on Question # 83830, Math / Functional Analysis

Question 1. Find a measure (of ?) dense subset (?) in $\mathbb{R}^2$ or

Find a measurable dense subset in $\mathbb{R}^2$?

Solution. $\mathbb{Q}^2$ is dense subset of $\mathbb{R}^2$ and $m(\mathbb{Q}^2) = 0$.

We construct an example of a dense open set of positive measure with the help of some subset $\mathcal{D}$. Let $\mathcal{D}$ be a dense, open subset of $\mathbb{R}^2$, whose Lebesgue measure is positive and finite. Define a function $f\colon \mathbb{R}^2 \to \mathbb{R}^2$ as $f(x,y) := \frac{\epsilon}{m(\mathcal{D})}(x,y)$, $\epsilon > 0$. Since $f$ is linear, $m(f(\mathcal{D})) = \frac{\epsilon}{m(\mathcal{D})} m(\mathcal{D}) = \epsilon$. Since $f$ is continuous, $f(\mathcal{D})$ is dense in $\mathbb{R}^2$. Since $f$ is invertible, and its inverse is again continuous, $f(\mathcal{D})$ is open.

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