I've encountered two statements regarding the Lebesgue measure that don't exactly contradict each other, but seem to me to be a little bit unintuitive when regarded with respect to one another.
The statements are:
- For each interval $[a,b]$
there exists a set $K\subset[a,b]$
that is compact , totally disconnected and with positive measure (i.e. doesn't contain an interval).- Let $A\subset\mathbb{R}$
be some measurable set, and $0\leq\alpha<1$
such that for every interval $I\subset\mathbb{R}$
it holds that $m(A\cap I)\leq\alpha m(I)$
then $m(A)=0$
The second statement seems to imply that any positive measure set must contain an interval up to a set of arbitrary measure $\epsilon>0$ (since a suitable $\alpha$ can't be found for a positive measure set).
On the other hand the first statement says that for any interval I can find a subset of any desired positive "not-full" measure.
I realize these statements don't contradict, but they seem a little "contradictory in nature". Intuitively I would expect the first statement to somehow allow me to construct a positive measure set with the property desired in the second statement.
I don't really have a question here but was hoping maybe someone can shed their own perspective on how they see these two properties of the Lebesgue measure co-existing. Maybe you can give me some further intuition.
Thanks a lot!
Best Answer
For 1: There is an open set $U$ containing all rationals such that $m(U)< (b-a).$ Thus $m([a,b]\setminus U) > 0.$ The set $[a,b]\setminus U$ is compact, is a subset of $[a,b],$ and since it contains no rational, has no interior.