Measure of Reals: Countable or Uncountable?

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In summary, the conversation discusses the concept of measure theory and the measure of a singleton, which is zero. The idea of describing the reals as an uncountable number of singletons with zero measure and taking their union is brought up, but it is noted that this principle only applies to countable sets. The question of how to define the sum of an uncountable set of numbers is raised, and the suggestion to do further research is given.
  • #1
cragar
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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.
 
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  • #2
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?
 
  • #3
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.
 

FAQ: Measure of Reals: Countable or Uncountable?

What is a measure of reals?

A measure of reals is a mathematical concept used to quantify the size or magnitude of a set of real numbers. It is a way to assign a numerical value to a set of numbers, representing the "size" of that set.

How is the measure of reals determined?

The measure of reals is determined using a mathematical tool called a measure function. This function takes in a set of real numbers and assigns a value to it based on certain criteria or properties.

What does it mean for a set of reals to be countable?

A set of reals is countable if it can be put into a one-to-one correspondence with the set of natural numbers (1, 2, 3, ...). This means that for every real number in the set, there is a unique natural number assigned to it, and vice versa.

What does it mean for a set of reals to be uncountable?

A set of reals is uncountable if it cannot be put into a one-to-one correspondence with the set of natural numbers. In other words, there is no way to assign a unique natural number to every real number in the set.

Can a set of reals be both countable and uncountable?

No, a set of reals can only be either countable or uncountable. This is because the definitions of countable and uncountable are mutually exclusive - a set cannot have both properties at the same time.

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