7
$\begingroup$

Show that for any $\delta>0$ there exists an open dense subset $U$ of $\mathbb{R}$ with $\mu(U)<\delta$, where the measure is the outer measure.

I'm not sure how to go here. The set $U$ must be dense (so for any point $x\in\mathbb{R}$ there exists a point in $U$ arbitrarily close to $x$), open (so any point in $U$ has a neighborhood entirely contained in $U$), and yet can be covered by a countable union of intervals with really small length.

$\endgroup$
12
$\begingroup$

$\mathbb{Q}$ is dense. Let $\{q_n : n \in \mathbb{N}\}$ enumerate $\mathbb{Q}$. Let $U = \bigcup_{n \in \mathbb{N}} B(q_n,\frac{\delta}{2^n})$. $U$ is open and has measure less than $\delta$. It is dense because $\mathbb{Q} \subseteq U$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.