Measure of Reals: Countable or Uncountable?

[cragar]

I was reading a little about measure theory, and the measure of a singleton is zero.
So why couldn't we just describe the reals as an uncountable number of singletons which each have zero measure and then union all of these singletons.
Maybe the union only works for countable sets when talking about measure.

 

[jbriggs444]

You are trying to apply the principle that the measure of a disjoint union is the sum of the measures of the components, right?

How do you define the sum of an uncountable set of numbers?

 

[micromass]

Literally looking 5 seconds at the wikipedia page on measure theory answers your questions: http://en.wikipedia.org/wiki/Measure_theory

Please do some research yourself before asking questions.