I am currently trying to solve some analysis exercises on metric spaces, but I cannot quite tackle on of them. The exercises read as follows:
Define the measure $$\mu_x:=\frac{d(\bullet,x)}{\mu(B(x,d(\bullet,x)))}\mu $$ where $B(x,R)$ denotes a ball centered at $x$ of radius $R$. We call this the Riesz potential centered at $x$.
Exercise 1: Assume $\mu$ is doubling. Show that $$\mu_x(B(x,R))\approx R$$ Exercise 2: Assume $\mu$ is doubling, $E$ Borel. Let $x$ be a Lebesgue point of $E$ wrt $\mu$; prove that $x$ is a Lebesgue point of $E$ wrt $\mu_x$.
I've been able to solve Exercise $1$ by cutting up the Ball into a sequence of annuli of decreasing radius and using the doubling property on those to bound the integral by above and below and get the stated result.
For Exercise $2$ I first thought that one could see this as a kind of Corollary to the first ( I've also tried to solve it analogously to the previous one), but all my tries have lead nowhere. Can someone give me a hint on how to start? And thus why do we want to assume the Borel property?
Many thanks in advance
Exercise 2 works similarly to the first using dyadic annuli centered at $x$. Let $$A_n=\{y: 2^{-n-1}\le d(x,y)<2^{-n}\}, \quad n=1,2,\dots$$ Since $x$ is a Lebesgue point of $E$ and $\mu$ is doubling, we have $$ \lim_{n\to\infty}\frac{\mu(A_n\setminus E)}{\mu(A_n)}= 0 \tag{1} $$ (Indeed, the denominator is comparable to $\mu(B(x,2^{-n}))$ by the doubling property.)
On the set $A_n$, the factor $d(y,x)/\mu(B(x,d(y,x)))$ is approximately constant, i.e., pinched between two positive constants whose ratio depends only on the doubling constant of $\mu$. This and (1) imply $$ \lim_{n\to\infty}\frac{\mu_x(A_n\setminus E)}{\mu_x(A_n)}= 0 \tag{2} $$ Property (2) implies that $x$ is a Lebesgue point of $E$ wrt $\mu_x$.