This is a very naive question. Consider a probability space $(\Omega, \mathcal{F},P)$ with the property that, for all $\omega \in \Omega$, $\{ \omega \} \in \mathcal{F}$ and $P(\{ \omega \}) = 0$. There are many familiar examples of such spaces (e.g. Lebesgue measure on $[0,1]$).

My question is

Is there an interesting way to characterize such spaces?

In addition to a characterization, I'd also be happy to learn of any conditions that are sufficient or necessary for this property to hold.


1 Answer 1


Partial answer: You first of all need the space ${\Omega}$ to be uncountable, or else having no atoms (i.e. every $P(\{w\})=0$) means that the total measure is zero (by countable additivity of the measure).

Under the continuum hypothesis you may assume that it is ${\Bbb R}$ (or e.g. an interval in ${\Bbb R}$). You need, however, also to have a $\sigma$-algebra of measurable subsets which allow for measures without atoms. The Borel $\sigma$-algebra is a good choice.


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