Consider the Cantor-like set $C$ resulting from the Cantor-like construction, which starts with $k$ disjoint closed intervals $\delta_i$, $k \ge 2$, $i=1,\dots,k$ of the unit interval. Given an $n$-tuple $(i_1\dots i_n)$, where $i_j=1,\dots , k$, let $\delta_{i_1 \dots i_n}$ be the corresponding interval on step $n$ of the construction and denote by $C_{i_1,\dots,i_n}=\delta_{i_1 \dots i_n} \cap C$. Let $E'$ be the collection of all sets $C_{i_1,\dots,i_n}$ over all $n$ and all $n$-tuples. Given a probability vector $P=\{p_1, \dots , p_{k}\}$ (where $p_i \ge 0$ and $p_1 + \cdots + p_{k} = 1$,) define a set function $\mu$ on $E'$ by the formula $$\mu(C_{i_1,\dots,i_n})=\prod_{j=1}^{n}p_{i_j}.$$If $\mu^*$ is the outer measure generated by $\mu$, and $\bar{\mu}$ the induced measure, show that every $C_{i_1,\dots,i_n}$ is measurable.

Let $A_{n_0} =C_{i_1,\dots,i_{n_0}}$, $E \subset [0,1]$. Note that $$\mu^*(A_{n_0} \cap E)=\inf_{\{C_{i_1,\dots,i_n}\}, n \in N}\{\sum_n \mu(C_{i_1,\dots,i_n}):A_{n_0} \cap E \subset \bigcup C_{i_1,\dots,i_n}\}.$$ So we may assume that $E$ contains no isolated points, as such points, if they were in $C$, can be covered by a sequence of nested intervals in $E'$ not containing other points of $E \cap C$, with measure $\to 0$. Thus every neighborhood of a point in $A_{n_0} \cap E$ contains a point in $A_{n_0} \cap E$. The question is, does this property necessarily imply that the only nontrivial $E$ are either (i) unions of intervals, or (ii) sets agreeing with $C$ at uncountably many points? If so, (i) is not difficult to solve. But what about (ii)? Can we somehow characterize the sets in (ii)?


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