Measure of Irrationals with Even First Digit

In summary, the conversation discusses concepts such as Lebesgue integration and Lebesgue measure of sets, specifically in regards to the irrationals on the interval [0,1]. The question is raised about the measure of a set A containing irrationals whose first digit in their decimal expansion is even. Intuitive thoughts suggest it should have a measure of 1/2, but it is difficult to cover these irrationals with open intervals of total length less than 1. The conversation also explores the possibility of dividing the irrationals from [0,1] into two sets, each containing "half" of the irrationals but with non-zero measures when intersected with any open interval. Finally, a possible solution
  • #1
Frillth
80
0
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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  • #2
Can you find any intervals in [0,1] that don't contain any such irrational numbers?
 
  • #3
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?
 
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  • #4
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
He wanted non-null, not non-empty.
 
  • #6
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?
 
  • #7
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)?
 

1. What is the Measure of Irrationals with Even First Digit?

The Measure of Irrationals with Even First Digit refers to the proportion of irrational numbers whose first digit is even out of all the irrational numbers in a given set or range. It is used to determine the frequency of occurrence of irrational numbers with even first digits.

2. How is the Measure of Irrationals with Even First Digit calculated?

The Measure of Irrationals with Even First Digit can be calculated by dividing the number of irrational numbers with even first digits by the total number of irrational numbers in a given set or range. The result is usually expressed as a percentage or decimal.

3. What is the significance of the Measure of Irrationals with Even First Digit?

The Measure of Irrationals with Even First Digit is important in understanding the distribution and patterns of irrational numbers. It can also help in identifying any biases or anomalies in a set of irrational numbers.

4. Can the Measure of Irrationals with Even First Digit be used for all irrational numbers?

Yes, the Measure of Irrationals with Even First Digit can be applied to all irrational numbers. However, it is more commonly used for numbers with infinite decimal expansions, such as pi and square root of 2.

5. How is the Measure of Irrationals with Even First Digit useful in real-world applications?

The Measure of Irrationals with Even First Digit can be used in various fields such as statistics, cryptography, and number theory. It can also help in analyzing and predicting patterns in financial data and stock market trends.

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