Construct a Cantor-type subset

Question

This problem is a part of an exercise in Zygmund's Real Analysis. It reads:

Construct a measurable set $E$ of $[0,1]$ such that for every sub-interval I, both sets $E\cap I$ and $I\setminus E$ have positive measure.

The hint is to consider Cantor-type subset, but if consider a Cantor-type set, there must exist a sub-interval I belong to E. So, it mean no interval should be in E. Any way if I belong to E,then $|E\cap I|<|E|=0$. So I can't really know how to use the hint.

Now my question is how can I modified it to met the condition?

The whole hint on book is "Take a Cantor-type subset of [0, 1] with positive measure, and on each subinterval of the complement of this sure sestruct another such set, and so on. The measures can be arranged so that the union of all the sets has the desired property."

Answer 1

This is a rather broad interpretation of "Cantor-type." The basic idea is the following (letting $Q_i$ be an enumeration of the countably many nontrivial open intervals with rational endpoints):

• We have countably many "positive requirements" $P_i\equiv m(E\cap Q_i)>0$. The obvious way to meet such a requirement is to put an interval into $E$.

• We have countably many "negative requirements" $N_i\equiv m(E\setminus Q_i)>0$. The obvious way to meet such a requirement is to throw an interval out of $E$.

The obvious issue is that the action we take to satisfy one requirement might "injure" another, e.g. putting $(0,1)$ into $E$ to make $E\cap (-3, 17)$ have positive measure obviously causes a problem for $E\setminus ({1\over 3}, {1\over 2})$. One solution is to allow injury but only to a very controlled extent; e.g. in the above example we might allow ourselves to put a very tiny subinterval of $({1\over 3}, {1\over 2})$ back into $E$. More abstractly, we'll want to keep track of some numerical parameter during our construction which tells us how much we're allowed to "alter" the set being built.

The details are, in my opinion at least, fun to work out for oneself, so I've spoilered the construction I have in mind:

We'll define a sequence of sets $E_i$ (= the stage-$i$ approximation to $E$), as well as a sequence of "stability parameters" $\epsilon_i$, as follows. We start with $E_0=\emptyset$ and $\epsilon_0={m(Q_0)\over 4}$. Next, having defined $E_i$ and $\epsilon_i$, we now face the pair of requirements $P_i$ and $N_i$. We pick a pair of disjoint open subintervals $I,J$ of $Q_i$ with $m(I)=m(J)=\epsilon_i$. Of course this requires $\epsilon_i$ to be "small" relative to $Q_i$; that will be true by induction, and it's with this in mind that we set $$\epsilon_{i+1}=\min\{{\epsilon_i\over 2^{i+2}}, {m(Q_{i+1})\over 2^{i+2}}\}.$$ More obviously, we also set $E_{i+1}=E_i\cup I\setminus J$. Finally, we take the "limsup" of the sequence of sets we've built so far: $$E=\{x:\forall m\exists n>m(x\in E_n)\}.$$ (The "liminf" would also work, the point is just that we need to account for oscillation of membership in the $E_i$s as $i\rightarrow\infty$.) Since the sequence $(\epsilon_i)_{i\in\mathbb{N}}$ shrinks sufficiently rapidly, and is always small relative to the interval being looked at at a given stage, it's easy to see that all requirements are satisfied.