I came up with the following intuitive explanation why the irrational numbers in the interval [0,1] have measure 1 and would like to know if the explanation is correct.
- the Lebesgue measure requires a countable union of disjoint
intervals to cover all irrationals in the interval [0,1] - the irrationals are uncountable
- Therefore, the (countable) intervals in the measure need to be of
non-zero length to cover all irrationals - since the whole interval needs to be covered the lengths sum to one
Any evaluation is appreciated
habbes
Best Answer
It is not clear what you mean by "the Lebesgue measure requires a countable union of disjoint intervals to cover all irrationals in the interval [0,1] ".
Let $A$ be the rational numbers in $[0,1]$ and $B$ be the irrational numbers in $[0,1]$.
Then $A$ and $B$ are disjoint and $A \cup B=[0,1]$, hence
$1= \lambda([0,1])= \lambda(A)+ \lambda(B)=\lambda(B)$,
since $\lambda(A)=0$.