What is the probability of a power set of the sample space?

Question

Let $\Omega$ be the sample space for an experiment, and, $F$ is the power set of $\Omega$.

We know that $P(\Omega)=1$.

What would be the probability of $F$?

That is, $P(F) = $?

Answer 1

We can only ask what the probability of a subset $A$ of $\Omega$ is. We can't answer this question because $F$ is the powerset of $\Omega$. It's not a subset of $\Omega$, so we can't ask what its probability is. (What I mean by "we can't" is that it doesn't make sense to ask this.)

Answer 2

For a simple case where you have $\Omega=\{H,T\}$, then $F=\{\emptyset,\{H\},\{T\},\{H\cup T\}\}$ and $P(F)=0+0.5+0.5+1=2$.

Similarly, for a three element set with each possibility $1\over3$, We will have $P(F)={1\over3}\times3+{2\over3}\times3+1=4$.

Start to see the pattern? Essentially, since $card(F)=2^{card(\Omega)}$ and any element appears in exactly in half of subsets of $F$, if we sum up (or integrate in case of infinite elements), each element will be added $2^{card(\Omega)}\over2$ times.

Hence what we are doing is essentially given $\int_{x\in\Omega}x=1$, we want to find $\int_{x\in\Omega}x({2^{card(\Omega)}\over2})$ which equals $2^{card(\Omega)}\over2$