Outer measure of the set $A= \lbrace x \in \mathbb{R}^n :\exists r>0 \; with \; \mu(B_r(x))=0 \rbrace$

Question

id like to show

if i have a outer measure $\mu$ and the set $A= \lbrace x \in \mathbb{R}^n :\exists r>0 \; with \; \mu(B_r(x))=0 \rbrace$ than $\mu(A) = 0$.

I don't know how to start. I need some help please.

Answer 1

Notice that, for $x\in\mathbb{R}^n$ and $r>0$, if $\mu(B_r(x)) = 0$, then $B_r(x)\subset A$ (why?).

I will make the following claim, which would prove your statement:

Claim: There exists a countable collection $\{x_i\}_{i\in Q}$ of points in $\mathbb{R}^n$ and a corresponding countable collection of positive numbers $\{r_i\}_{i\in Q}$ such that $A\subset\cup_{i\in Q}{B_{r_i}(x_i)}$ and $\mu(B_{r_i}(x_i)) = 0$ for every $i\in Q$.

Proof: Take a countable dense subset $D$ of $\mathbb{R}^n$ (e.g. $D = \mathbb{Q}^n$), and consider the intersection $A' = A\cap D$. Then $A'$ is countable, so we can write $A' = \{x_i\}_{i\in Q}$ for some countable index set $Q$. For each $i\in Q$, the fact that $x_i\in A'\subset A$ implies that there exists some $r>0$ with $\mu(B_r(x_i)) = 0$. For each $i\in Q$, let $$ r_i = \sup\{r>0\,|\,\mu(B_r(x_i)) = 0\}.$$ Notice that if $r_i = \infty$ for any $i$ then $\mu(\mathbb{R}^n) = 0$, and we would trivially be done. Otherwise, $0<r_i<\infty$ for every $i$, and in fact we have $\mu(B_{r_i}(x_i)) = 0$ for all $i$ (why?). So it suffices to show that $A\subset\cup_{i\in Q}{B_{r_i}(x_i)}$.

To show this, let $x\in A$. Then there exists $r>0$ such that $\mu(B_r(x)) = 0$. By density of $D$, there exists some $y\in D$ such that $y\in B_{r/2}(x)$. Then $y\in A'$ (why?), so it can be written as $y = x_i$ for some $i\in Q$, and I leave it as an exercise to you to prove that for this $i$ we have $x\in B_{r_i}(x_i)$.