Let us assume that we have a sample space $\Omega=[0,1]$. Why is it not possible to have all the singletons $\{x\}\in\Omega$ with non-zero probability measure.
I searched for an answer on this forum but I could not find an answer. Please point me to a thread which gives an answer or please let me know as to why all the singletons of an uncountable sample space can never have non-zero probability measure. I know that only a countable number of them can have a non-zero probability measure but I do not have a proof for that.
Best Answer
Assume this was the case. Consider the following partition of $\Omega$ into countably many sets: $$ A_n = \left\{x \in \Omega : 1/(n+1)\leq \mathbb{P}(\{x\}) \lt 1/n)\ \right\} $$ Then, this forms a disjoint union of $\Omega$, so that: $$ \mathbb{P}(\Omega) = \sum_{ n =1}^\infty \mathbb{P}(A_n) = 1 $$ Now, for every singleton the set, $\mathbb{P}(x) \geq 1/(n+1)$ for some $n$. Fix $n$ and consider $A_n$. If this set had infinitely many (say countably many elements), then already we would have that $\mathbb{P}(A_n) = \infty$ as we would be summing $\sum_{k = 1}^\infty 1/(n+1)$ (here $n$ is fixed). Thus, each set has finitely many elements, and so $\Omega$ is a countable union of finite sets, ergo countable.
Note that the proof by contradiction was unnecessary here, but I have kept it for the sake of clarity.