Measure-theoretic boundary of a fat Cantor set

Question

Let $C_\lambda\subset [0,1]$ be the fat Cantor set of parameter $\lambda$ (which is constructed as the usual Cantor set, removing at the $n$-th step the middle intervals of length $\lambda / 3^n$).

It is well-known that $C_\lambda$ is a perfect set, hence its topological boundary $\partial C_\lambda$ coincides with $C_\lambda$.

On the other hand, I am not able to figure out what is the measure-theoretic boundary of $C_\lambda$. (I recall that the measure-theoretic boundary $\partial^* E$ of a measurable set $E$ is the set of points with density different from $0$ and $1$.)

Any hint or reference will be appreciated.

You are right: indeed $C_\lambda$ should have measure $1-\lambda$. At the $n$-th step the removed intervals have measure $\lambda / 3^n$. — Rigel, Oct 06 '18 at 09:52
the measure-theoretic boundary $\partial^* E$ of a measurable set $E$ is the set of points with density different from $0$ and $1$ --- Do you mean points where the density EXISTS and is different from $0$ and $1,$ or do you mean points at which the density does not equal either $0$ or $1$? (The 2nd version includes points at which the density does not exist.) Anyway, what follows will probably not help much, but for what it's worth, given any closed and nowhere dense set $C,$ the Baire-typical point in $C$ (Baire category relative to the space $C)$ (continued) — Dave L. Renfro, Oct 06 '18 at 16:16
has both unilateral densities equal to zero (follows immediately from a theorem I called "Denjoy's Theorem" in this paper), and thus for either version of your definition I offered, the set you're looking for will be a Lebesgue measure zero set that is meager relative to $C.$ — Dave L. Renfro, Oct 06 '18 at 16:18
On the off-chance they could be of use (or even just of interest to you), see Specific example case for Lebesgue density *AND* Lebesgue points of density and similar notions *AND* For every null set $E$ there is a measurable set $F$ with different upper and lower Lebesgue density at every point of $E$ *AND* For Every Measure Zero Set $E$ There Exists a Positive Measure with $\cdots$. — Dave L. Renfro, Oct 06 '18 at 17:41

Measure-theoretic boundary of a fat Cantor set

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