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Q. Construct infinitely many disjoint sets $A_1, A_2,... \subset R$, each of which is a union of suitable symmetric Cantor sets, such that for every interval I and every $k=1,2,...$ the intersection $A_k \cap I$ has positive length.

I'm really struggling with this.The cantor sets are meagre on the interval they are defined so you can easily define some interval that will have empty intersection with any cantor set. Then assuming that we fill the gaps of a cantor set by taking unions with other cantor sets I can't see how to make an infinite family of sets that are a union of cantor like sets but still remain pairwise disjoint.

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@t.b.: It’s not quite a duplicate, since a small extra trick is required to get infinitely many sets instead of one. –  Brian M. Scott Oct 19 '11 at 20:16
    
I dont quite see how you could the given solution to answer this question :( especially even if we find cantor sets satifying the above conditions how do we ensure that each A_i intersects with every interval in R? –  user17957 Oct 20 '11 at 18:57
    
I've got as far as defining $A_1$ with the desired properties but I'm struggling now with how to extend this to an infinite family. I'm thinking of a translation of some sort. –  user17904 Oct 20 '11 at 19:20
    
I was thinking of defining the A_i 's as 'union of cantor sets' over unit intervals (i.e pick a unit interval , define a cantor set on it hich has positive measure , and then fill in the 'gaps' on the interval which are not occupied by cantor sets with more and more cantor sets of positive measure and sort of cover de whole unit interval with cantor sets) so doing this way we can have an infite sequence disjoint sets A_i, with each i corresponding to a unit interval. The problem I'm having with this approach is that there is nothing to ensure that each interval in R will have a non-empty inter –  user17957 Oct 20 '11 at 20:10

1 Answer 1

Enumerate all open intervals with rational coordinates as $(I_j)_{j\ge 1}$. Enumerate all pairs $(k,j)_{k,j\ge 1}$ as $(k(n),j(n))_{n\ge 1}$. For each $n$ choose a symmetric Cantor set of positive measure $C_n$ such that $$C_n\subset I_{j(n)} \setminus \bigcup_{m=1}^{n-1}C_m \tag1$$ This is possible because the set on the right in (1) is open and nonempty (even comeager). The sets $$A_k=\bigcup_{k(n)=k} C_n,\quad k\ge 1$$ meet the requirements.

The above is a slight adaptation of Construction of a Borel set with positive but not full measure in each interval.

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