Intuitive explanation why Lebesgue measure of irrationals in [0,1] equals 1

Question

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

Answer 1

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 $\lambda(A)=0$.