Measure of Irrationals with Even First Digit

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Discussion Overview

The discussion revolves around the measure of the set of irrationals in the interval [0,1] that have an even first digit in their decimal expansion. Participants explore the implications of this measure, particularly whether it could be 1/2 or 1, and how this relates to the overall measure of irrationals in that interval.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the measure of the set A (irrationals with even first digit) should intuitively be 1/2, as it seems to represent "half" of the irrationals in [0,1].
  • Another participant questions how to cover the irrationals in A with open intervals of total length less than 1, implying a challenge to the measure being 1/2.
  • A new question is posed about whether it is possible to divide the irrationals into two sets, A and B, such that both sets contain "half" of the irrationals while maintaining non-zero measure in any open interval contained in [0,1].
  • One participant suggests using dense null sets to create two sets with the desired properties, but this is challenged by another participant who clarifies that non-null sets are required.
  • There is a discussion about specific intervals, such as [.11, .12], that do not contain any irrationals starting with an even digit, raising questions about the distribution of these irrationals.

Areas of Agreement / Disagreement

Participants express differing views on the measure of the set A, with some supporting the idea of it being 1/2 and others questioning this assumption. The discussion remains unresolved regarding the measure and the feasibility of dividing the irrationals as proposed.

Contextual Notes

There are limitations regarding the assumptions made about the distribution of irrationals and the properties of measures in this context. The discussion does not resolve how to definitively prove the measure of set A or the implications of dividing the irrationals.

Frillth
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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!
 
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Can you find any intervals in [0,1] that don't contain any such irrational numbers?
 
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:
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)
 
He wanted non-null, not non-empty.
 
Hurkyl said:
Can you find any intervals in [0,1] that don't contain any such irrational numbers?

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?
 
frillth said:
however, i can't think of any way to cover these irrationals with open intervals of any total length less than 1.

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

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