Understanding Measure Zero and Countable Additivity for Rational Numbers

[yifli]

A book I'm reading says:

If a set A has infinitely many points which can be arranged in a sequence $a_1,a_2,\cdots,$, then A has measure zero.

What does it mean by "can be arranged in a sequence"? The book gives an example on the set A which is all the rational numbers between 0 and 1. Why can't irrational numbers be arranged in a sequence?

 

[spamiam]

A book I'm reading says:

If a set A has infinitely many points which can be arranged in a sequence $a_1,a_2,\cdots,$, then A has measure zero.

What does it mean by "can be arranged in a sequence"? The book gives an example on the set A which is all the rational numbers between 0 and 1. Why can't irrational numbers be arranged in a sequence?

I think the point is that a sequence is indexed by the positive integers, so the set of all members of the sequence is countable. The rationals are countable, so we can put them into a one-to-one correspondence with the positive integers, giving us our sequence. The irrationals are uncountable, so we can't do the same thing with them. Bottom line: countable sets have measure zero.

Is this from Royden? I remember he did strange things with sequences, when really countability is just about a bijection with the positive integers.

 

[pwsnafu]

One of the axioms for measures is countable additivity: so if A is a sequence a₁, a₂, ... then

μ(A) = μ({a₁}) + μ({a₂}) + ...

If my measure is zero on individual points, then any countable set also has zero measure.