Infinite Sample Space I came across this line in a textbook I'm reading,

When $\Omega$ is infinite, its power set is too large a collection for probabilities to be assigned reasonably to all its members.

I'm not quite able to wrap by head around this. Probability is a positive function from a collection of events to $[0,1]$. The disjoint additivity property also holds. 
Now both $[0,1]$ and $\{0,1\}^\Omega$ are uncountably infinite so why can't there exist a map between the two? An intuitive explanation would help a lot!
 A: When one assigns a probability measure, one would like it to have some reasonable properties:


*

*$P( \cup_i A_i) = \sum_i P(A_1)$

*$P(A + x) = P(A)$ (translation preserves measure)

*$P(\emptyset) = 0$

*$P(\Omega) = 1$
Consider now $\Omega = [0,1]$ And as a sigma algebra one considers $\mathcal{P}(\Omega) = \{0,1\}^{\Omega} = \{A \mid A \subset \Omega\}$
Now consider the following equivalence relations: 
We say that $x \sim y $ when $\exists\, q \in \mathcal{Q} $ such that $x = y + q$ 
Consider the class $[x] = {y \in [0,1]: y \sim x}$ now we can decompose $[0,1] = \cup_{x\in[0,1]} [x]$. Consider now the set $A_0$ to be made from a point from each distinct $[x]$ this is possible only if we use the axiom of choice (since we are chosing a point of infinite non- empty sets)
Let $A_q = \{x+q \text{ (mod 1)}\mid x \in A_0\}$ note that $A_{q_1} \cap A_{q_2}  = \emptyset$ if $q_1 \neq q_2$ ($q_1 -q_2 \neq 0 $ (mod 1)) and that $\cup_q{A_q} = [0,1]$
So now what is the value of $P(A_q)$
$P(A_q) = P(A_0 + q) = P(A_0)$
$1 = P([0,1]) = P(\cup_q A_q) = \sum_q P(A_q) = \infty P(A_0)$
therefore $P(A_0)$  can't be zero and can't be non zero,
This set is a strange set. We use the notion of measurability to avoid such pathologies, take a look at
https://en.wikipedia.org/wiki/Non-measurable_set
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Hope this helps
