Equivalently, is it possible for uncountable number of positives to sum to $1$?
I was thinking about bounding some how the numbers to prove that it is not possible (I have a feeling that it is not possible, I haven't encountered a random variable that takes uncountably many values with positive probability). Now suppose that there exists a random variable $X$ that takes all values in $[0,1]$ with positive probability (because any uncountable set can be mapped to $[0,1]$). Now, pick the maximum of the probabilities, then pick second maximum and so on and sum them. If that is more than $1$ we get a contradiction. But there are many problems with this greedy approach:
- We are going along a sequence, that can be summed to $1$, even if all the probabilities can't be.
- Uncountable sets are not enumerable by definition.
How do I go about proving it?
