Basic measurement theory question

[BWV]

Trying to get my head around some basic points of measure theory

So rational numbers are dense in the reals. I.e., if

with

, then there exists an

such that

. It follows that there are then infinitely many such.

The Lebesgue measure of any single irrational (or rational number) is zero in $\mathbb{R}$ or $\mathbb{Q}$

let x = an irrational number, say √2
Let s=set of rational numbers approximating x to i decimal places with i = [0,∞)

What is the Lebesgue measure m(s)?

also, if m($\mathbb{R}$)>m($\mathbb{Q}$), how do you account for the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$, i.e. for every irrational number, there are an infinite number of rational approximations (s above)?

 

[Office_Shredder]

Do you know what $m(\mathbb{Q})$ is? If so you should be able to compute m(s) easily.

There are an equal number of integer points as there are rational points. There are *so* many more real numbers. With that said, cardinal size of the set doesn't say much about the measure, but I think that's a good starting point for the intuition here.

One last thing to note, $m(\mathbb{R})$ doesn't exist (or is infinity)

 

[BWV]

Do you know what $m(\mathbb{Q})$ is? If so you should be able to compute m(s) easily.

There are an equal number of integer points as there are rational points. There are *so* many more real numbers. With that said, cardinal size of the set doesn't say much about the measure, but I think that's a good starting point for the intuition here.

One last thing to note, $m(\mathbb{R})$ doesn't exist (or is infinity)

Ok thanks, still getting my head around it, but the measure of any countable set is zero and so is my example s above

 

[FactChecker]

i.e. for every irrational number, there are an infinite number of rational approximations (s above)?

Only if you reuse the rational numbers a lot to get close to all those irrational numbers. Are you familiar with the fact that the set of rational numbers is countably infinite and that the set of irrational numbers is uncountably infinite? Since the rational numbers are countable, you can enclose each of them in a sequence of smaller and smaller intervals where the total length of the intervals is as small as you want. That proves that the measure of the rationals is zero. measure(rationals in [0,1])= 0. You can not do the same thing with the irrationals because they are uncountable and you can not sum the interval lengths of an uncountable number of intervals. Furthermore, the irrationals in [0,1] have measure 1-measure(rationals in [0,1]) = 1-0 = 1.

 

[BWV]

I am reading this in the context of statistics, with the Lebesgue measure tying to probability, and to do this everything is mapped to the reals, so a distribution that only takes integer values would have to be mapped with a step function? https://en.wikipedia.org/wiki/Step_function.

Read some of the definitions, but a reasonable informal definition of why the reals are uncountable is that you cannot distinguish between any number of irrational numbers that may have the same digits up to some arbitrarily large number of decimal places?

 

[FactChecker]

I am reading this in the context of statistics, with the Lebesgue measure tying to probability, and to do this everything is mapped to the reals, so a distribution that only takes integer values would have to be mapped with a step function? https://en.wikipedia.org/wiki/Step_function.

Yes. Probability is one of the common applications of Lebesgue measures.

Read some of the definitions, but a reasonable informal definition of why the reals are uncountable is that you cannot distinguish between any number of irrational numbers that may have the same digits up to some arbitrarily large number of decimal places?

The best formal or informal way to see that the real numbers are uncountable (interpret uncountable as un-listable) is the Canter diagonal proof. It is a proof by contradiction. If you imagine that you have listed ALL the real numbers in [0,1], then it is still easy to show one (in fact incredibly many) that you have missed from your list.