A Correspondence Theorem

The cardinality of the collection of uncountable closed subsets of $\mathbb {R}$ is $\mathrm{c}$.

Let $\omega$ denote the smallest uncountable ordinal number. From (a), we can form a well-ordering $\left\{ F_{\eta }: \eta \prec \omega \right\}$ of the uncountable closed subsets of $\mathbb {R}$.

Let $x_{0}$ and $y_{0}$ denote two distinct points of $F_{0}$. Observe that $A_{1}=\left\{ x_{0}, y_{0}\right\}$ has cardinal number less than $\mathrm{c}$, hence $\operatorname {card}\left(F_{1}-A_{1}\right)=\mathrm{c}$. Therefore, we may select distinct $x_{1}$ and $y_{1}$ from $F_{1}-A_{1}$. Observe that $A_{2}=\left\{ x_{0}, y_{0}, x_{1}, y_{1}\right\}$ has cardinal number less than $\mathrm{c}$, hence $\operatorname {card}\left(F_{2}-A_{2}\right)=\mathrm{c}$. Pick two distinct points $x_{2}, y_{2} \in F_{2}-A_{2}$, and continue. Employ the following transfinite recursion scheme: let $\eta \prec \omega$ denote an infinite ordinal, and suppose that $x_{\gamma }$ and $y_{\gamma }$ have been defined for all $\gamma \prec \eta$. The set $A_{\eta }=\left\{ x_{\gamma }: \gamma \prec \eta \right\} \cup \left\{ y_{\gamma }: \gamma \prec \eta \right\}$ has cardinality less than $\mathrm{c}$ [prove!], and thus $\operatorname {card}\left(F_{\eta }-A_{\eta }\right)=\mathrm{c}$, hence we may choose distinct $x_{\eta }, y_{\eta } \in F_{\eta }-A_{\eta }$. Let $B=\left\{ x_{\eta }: \eta \prec \omega \right\}$. Then $B \cap F_{\eta }$ and $B^{c} \cap F_{\eta }$ are nonempty for each $\eta \prec \omega$.

Let $F \in \mathfrak {S}$ be continuous and nonconstant. Then $\Delta_{F}(A)>0$ for some $A=(a, b]$.

Let $F \in \mathfrak {S}$ be continuous and nonconstant, so that $\Delta_{F}$ and $\Delta_{F}^{*}$ are nontrivial [that is, they are not identically zero]. If $B \in \mathcal{M}\left(\Delta_{F}^{*}\right)$, then

where $\Delta_{F, \mathcal{M}\left(\Delta_{F}^{*}\right)}^{*}$ is the restriction of the outer measure $\Delta_{F}^{*}$ to $\mathcal{M}\left(\Delta_{F}^{*}\right)$.

Given how $B$ was defined, the only compact subsets of $B$ are amc.

For each $x \in \mathbb {R}$, $\Delta_{F, \mathcal{M}\left(\Delta_{F}^{*}\right)}^{*}\left(\left\{ x\right\} \right)=0$. [$F$ is of course continuous in this and all subsequent parts.]

By (e)-(g), $\Delta_{F, \mathcal{M}\left(\Delta_{F}^{*}\right)}^{*}(B)=0$.

$\Delta_{F, \mathcal{M}\left(\Delta_{F}^{*}\right)}^{*}\left(B^{c}\right)=0$.

The results of (h) and (i) force $F$ to be constant, and hence a contradiction ensues. Conclude that $B \notin \mathcal{M}\left(\Delta_{F}^{*}\right)$ for any continuous nonconstant $F \in \mathfrak {S}$.

There exists an indexing $\left\{ C_{\eta }: \eta \prec \omega \right\}$ of the collection of nowhere dense closed subsets of $[0,1]$, where $\omega$ again denotes the smallest uncountable ordinal.

Employ a transfinitely recursive scheme to obtain a set $\left\{ x_{\eta }: \eta \prec \omega \right\}$ by the following: let $x_{0} \in [0,1]-C_{0}$; suppose that for each $\eta \prec \omega$ the number $x_{\gamma }$ is defined for each $\gamma \prec \eta$. Pick $x_{\eta } \in [0,1]-\bigcup_{\gamma \prec \eta }\left(C_{\gamma } \cup \left\{ x_{\gamma }\right\} \right)$. This construction requires (AC). [Precisely where?]

Let $S=\left\{ x_{\eta }: \eta \prec \omega \right\}$. Then (CH) forces $\operatorname {card}(S)=\mathrm{c}$.

For every $\eta \prec \omega$, we have that $S \cap C_{\eta }$ is amc.

$\Delta_{F, \mathcal{M}\left(\Delta_{F}^{*}\right)}^{*}(S)=0$ for every continuous $F \in \mathfrak {S}$.