Intuition of subsets which are not in the sigma algebra I am studying probability theory and from what I have understood is that when the sample space is uncountable, the probability measure cannot assign probabilities to every possible subset of the sample space, hence we build another set containing the subsets of the sample space to which we can consistently assign probabilities to.
But I was wondering about those other subsets which are left out and not included in the sigma-algebra. Are those probabilities 0 or do we not even consider those subsets as events? For example, the probability of choosing a number 1/2 between [0,1] is 0. Here is the event of choosing the number 1/2 defined to be equal to 0, or do we not even consider that to be an event?
Any help or intuition would help a lot. Thanks in advance.
 A: You have to be a bit more careful. We can define a probability measure on the power set of an uncountable set. For instance, with $x\in[0,1]$, the Dirac measure $\delta_x(A):=1$ if $x\in A$ and $\delta_x(A)=0$ otherwise can be defined on the power set of $[0,1]$. It is only certain properties we might want to enforce in addition to the defining properties of a measure which might force us to exclude some sets. In particular, there can be no measure $\lambda$ on the power set of $[0,1]$ which assigns each interval $[a,b]\subseteq[0,1]$ its length $\lambda([a,b])=b-a$, because we can construct sets $V$ ($V$ for Vitali, the guy who first constructed these) where any choice for $\lambda(V)$ leads to inconsistencies. But this only applies to this special choice of a measure and related measures. Not all measures!
As for the sets we exclude: we simply do not consider them events, so we don't assign any probability to them. In particular, we also don't say that they have probability $0$. We don't talk about them at all. If an event has probability $0$ it is still an event and part of the $\sigma$-algebra.
