Measure of Irrationals with Even First Digit

1. May 12, 2010

Frillth

I just finished a course where we discussed concepts such as Lebesgue integration and Lebesgue measure of sets. Today, I was telling my brother about how the irrationals on the interval [0,1] have measure 1, which is sort of counter-intuitive.

Anyway, he proposed the following question. Let A be the set of irrationals on the interval [0,1] whose first digit in their decimal expansion is even. What is the measure of A? Intuitively, I feel like it should have measure 1/2, since it should capture "half" of the irrationals on [0,1]. However, I can't think of any way to cover these irrationals with open intervals of any total length less than 1.

So if A does have measure 1/2, how can we prove that? If A has measure 1, then how do we reconcile this with the fact that the measure of the irrationals on [0,1] is 1?

Thanks!

2. May 12, 2010

Hurkyl

Staff Emeritus
Can you find any intervals in [0,1] that don't contain any such irrational numbers?

3. May 12, 2010

Frillth

Hahaha, of course. Man, it is clear that I haven't slept for a while. Thanks, Hurkyl.

Edit: Well, since my first question was pretty stupid, I have a new one to ask. Is it possible to divide up the irrationals from [0,1] into two sets A and B such that each set contains "half" of the irrationals, but for any open interval O contained in [0,1], m(OnA) and m(OnB) are both non-zero?

Last edited: May 12, 2010
4. May 12, 2010

Office_Shredder

Staff Emeritus
Start by taking two dense null sets (the rationals and an irrational translate) then just split the rest of the reals up between the two sets using whatever measure 1/2 breakdown you want (like the one described here)

5. May 12, 2010

Hurkyl

Staff Emeritus
He wanted non-null, not non-empty.

6. May 13, 2010

NaturePaper

Surely I am missing some simple points, but what about say [.11, .12]? Every real within this interval must start with 1 and so does not contain any such number?

7. May 16, 2010

daveyp225

(0,0.1)u(0.2,0.3)u(0.4,0.5)u(0.6,0.7)u(0.8,0.9)?