关于加倍度量空间的命题证明及相关逆问题的详细解答请求

基础定义：加倍度量空间 #

A metric space $(X,d)$ is said to be doubling if there exists a constant integer $N\ge 1$ such that every closed ball of radius $R$ can be covered by at most $N$ closed balls of radius $R/2$.

待证明的命题 #

Proposition
If a metric space $(X,d)$ is doubling, then there exist constants $C\ge 1, s>0$ such that for every $x\in X$, $0<r<R$ it holds the following implication:

If $Y$ is a subset of the ball $\mathbb{B}(x,R)$ such that $d(y,y')>r$ for every $y,y'\in Y, y\ne y'$, then $Y$ has cardinality less or equal than $C(R/r)^s$.

我的两个问题 #

问题1：上述命题的证明方法 #

How can I prove the Proposition? My idea is to use the following remark:

Every doubling metric space with doubling constant $N$ has de Groot dimension $N$, that is to say, for each $x\in X, r>0$, it holds that the cardinality of $Y$ is less or equal than $N$ for every set $Y\subseteq\mathbb{B}(x,r)$ such that $d(y,y')>r$ for every $y,y'\in Y$ with $y\ne y'$.

问题2：逆命题的证明 #

How can I prove that if there exists a doubling Borel outer measure over $(X,d)$, then the metric space is doubling?

I know that there exists a similar question to this on MathStack, but I'm looking for a more complete and detailed explanation. Can anyone help me please?

备注：内容来源于stack exchange，提问作者Grace53