- #1
greypilgrim
- 547
- 38
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"?
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"?