Generalization of measure theory to uncountable unions

In summary, the conversation discusses the possibility of a measure theory for uncountable unions and the formulation of an additivity property. It is mentioned that only the trivial measure can satisfy this when considering all uncountable unions on ##P(\mathbb{R})##. The conversation then moves on to the question of whether there can be a nontrivial subset of uncountable unions that allows for a nontrivial "measure." The conversation also touches on the use of hyperreal-valued measures and the difficulty of interpreting a set with a non-zero non-standard part having a measure. Finally, the conversation discusses the possibility of a partition of the unit interval into an uncountable number of subsets, all with nonzero Lebesgue
  • #1
greypilgrim
508
36
Hi.

Is there some kind of measure theory generalized to uncountable unions? Of course one needs to take care how to make sense of sums over an uncountable index set. I was thinking about following formulation of the additivity property of the "measure":
$$\mu\left(\bigcup_{i\in\ I} A_i\right)=\sup_{J\subset\ I}\sum_{i\in\ J}\mu\left(A_i\right)$$
Here ##I## can be uncountable, but the supremum is taken over all countable subsets ##J##.

Obviously, if we allow all uncountable unions on ##P(\mathbb{R})##, only the trivial measure can satisfy this (because we can divide every set into subsets of single elements with "measure" 0). But can there be a (nontrivial) subset of uncountable unions that allows for a nontrivial "measure"?
 
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  • #2
greypilgrim said:
Hi.

Is there some kind of measure theory generalized to uncountable unions? Of course one needs to take care how to make sense of sums over an uncountable index set. I was thinking about following formulation of the additivity property of the "measure":
$$\mu\left(\bigcup_{i\in\ I} A_i\right)=\sup_{J\subset\ I}\sum_{i\in\ J}\mu\left(A_i\right)$$
Here ##I## can be uncountable, but the supremum is taken over all countable subsets ##J##.

Obviously, if we allow all uncountable unions on ##P(\mathbb{R})##, only the trivial measure can satisfy this (because we can divide every set into subsets of single elements with "measure" 0). But can there be a (nontrivial) subset of uncountable unions that allows for a nontrivial "measure"?

I am not sure, sorry, but I have thought of using hyperreal-valued measures, since this would allow for sums to assume infinite values. Still, it seems hard to interpret a set having a measure with non-zero non-standard part. I think I saw this dealt with somewhere; let me see if I can find a link to the source. Your taking limits over countable subsets kind of reminds me of Riemann sums.
 
  • #3
Maybe I should have started with a more concrete question: Is there a partition of the unit interval into an uncountable number of subsets that all have nonzero Lebesgue measure?
 
  • #4
greypilgrim said:
Maybe I should have started with a more concrete question: Is there a partition of the unit interval into an uncountable number of subsets that all have nonzero Lebesgue measure?
If I understood correctly, no, if by partition you mean breaking down into pairwise-disjoint subsets.. For sum to converge ( meaning to be finite in this case) , support must be countable ; finite or infinite, of course.
 
  • #5
greypilgrim said:
Maybe I should have started with a more concrete question: Is there a partition of the unit interval into an uncountable number of subsets that all have nonzero Lebesgue measure?
No, because then, there must be an infinite (in fact uncountable) set of sets from the partition which all have measure > 1/n for some specific n > 0 (otherwise the partition would be countable, since it would be a countable union of countable sets), and then the measure of the unit interval would be infinite, a contradiction.

I think this is what WWGD meant.
 
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  • #6
Erland said:
No, because then, there must be an infinite (in fact uncountable) set of sets from the partition which all have measure > 1/n for some specific n > 0 (otherwise the partition would be countable, since it would be a countable union of countable sets), and then the measure of the unit interval would be infinite, a contradiction.

I think this is what WWGD meant.
Yes; exactly what I meant define ##S_n:=\{ s: s>1/n\} ## for ##s## in the collection . Then , by cardinality argument, at least one such ## S_n## is infinite...
 
Last edited:

1. What is measure theory and why is it important?

Measure theory is a branch of mathematics that deals with the concept of measuring the size or extent of sets. It provides a rigorous framework for assigning a numerical value (measure) to subsets of a given set. This theory is important in various areas of mathematics, such as probability, analysis, and geometry.

2. What does it mean to generalize measure theory to uncountable unions?

Uncountable unions refer to the combination of an infinite number of sets that cannot be put into a one-to-one correspondence with the natural numbers. Generalizing measure theory to uncountable unions involves extending the concepts and principles of measure theory to these types of unions, which allows for a more comprehensive understanding of the size and properties of these sets.

3. What are some applications of generalizing measure theory to uncountable unions?

Generalizing measure theory to uncountable unions has various applications in mathematics, particularly in the areas of integration, probability, and functional analysis. It also has practical applications in fields such as economics, physics, and engineering, where the concept of uncountable unions is often encountered.

4. What are the challenges in generalizing measure theory to uncountable unions?

One of the main challenges in generalizing measure theory to uncountable unions is dealing with the concept of uncountable additivity, which states that the measure of a countable union of disjoint sets is equal to the sum of their individual measures. This concept becomes more complicated when dealing with uncountable unions, as there may be an infinite number of sets to consider.

5. How is the generalization of measure theory to uncountable unions related to the concept of Lebesgue measure?

The generalization of measure theory to uncountable unions is closely related to the concept of Lebesgue measure, which is a measure defined on the real line that extends the notion of length to higher dimensions. The Lebesgue measure can be seen as a special case of the generalization of measure theory to uncountable unions, where the sets being measured are intervals on the real line.

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