Excising a compact subset of a set of finite Lebesgue measure

Question

We know from measure theory that for every $\epsilon>0$, we can always find a compact subset $K$ of a Lebesgue measurable set $E \subset \mathbb{R}$ of finite measure such that $m(E - K) \leq \epsilon$. My question is, it possible for us to do a similar approximation if we impose that $K$ is a finite union of disjoint compact intervals?

Answer 1

Yes or no, depending on how "similar" you require the approximation to be.

No: As already noted, if $m(E)>0$ and $E$ has empty interior then you certainly cannot approximate $E$ from the inside by a finite union of intervals.

Yes: But we should note that if $m(E)<\infty$ and $\epsilon>0$ there does exist a finite union of intervals $J$ with $$m(E\Delta J)<\epsilon,$$where $$A\Delta B=(A\setminus B)\cup(B\setminus A)=(A\cup B)\setminus (A\setminus B)$$is the set of points in one of $A$ and $B$ but not both.

(Proof: Take $K\subset E$ with $K$ compact and $m(E\setminus K)<\epsilon/2$. Say $K\subset V$ with $V$ open and $m(V\setminus K)<\epsilon.$ Now $V$ is a union of open intervals and $K$ is compact...)

Answer 2

A fat Cantor set does not contain any open interval. So this kind of approximation is not possible.

[All the intervals in $K$ have to be degenerate intervals so the measure of $K$ must be $0$].

Ref. for fat Cantor sets: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set