Suppose we have a closed subset $A\subset[0,1]$ that is not equal to $[0,1]$. Is it possible $mA=1$? Suppose you have an open subset $B\subset[0,1]$ that is dense in $[0,1]$. Is it possible that $mB<1$? Where $mA$ is the Lebesgue measure of some set $A$ of real numbers.
I've spent the last 4 or so hours thinking about what what the possibilities are here. Rationals and Irrationals and Cantor sets oh my!!! Truth be told, I would love to have an intuitive feeling for the intricate structure of all the real numbers in $[0,1]$ but alas I feel I am no closer to this goal than when I started looking at it a couple years ago.
When I look at these measurable sets problems Im always trying to construct examples in my head or to construct some sketches in an effort to intuit whats going on but often run into the brick wall that is the actual complexity of the real numbers. Anyone have a suggestion on how to build intuition here?