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On the axioms of probability we have that on the one hand the probability of the sample space $\Omega$ to happen is 1; in other words: $$ \mathbb{P}(\Omega) = 1$$

But on the other hand we also have that for any probability space the probability of the empty set is 0; in other words: $$ \mathbb{P}(\emptyset) = 0$$

So my question is, what if we choose our sample space $\Omega$ to be $\Omega = \emptyset$? What's the probability then?

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This seeming contradiction is simply a proof that there exists no probability space for which $\Omega=\emptyset$. A priori, given a set $\Omega$ and a $\sigma$-algebra $\Sigma$ on $\Omega$, there can be any number of different measures $\mathbb{P}$ which make $(\Omega,\Sigma,\mathbb{P})$ satisfy the axioms of a probability space--usually there are many different possible probability measures $\mathbb{P}$, but there might not be any at all! In the case that $\Omega=\emptyset$, the only possible $\sigma$-algebra on $\Omega$ is $\Sigma=\{\emptyset\}$, and there is no function $\mathbb{P}:\Sigma\to [0,1]$ which satisfies the axioms of a probability measure (since the axioms force $\mathbb{P}(\emptyset)$ to be both $0$ and $1$).

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  • $\begingroup$ Thanks Eric, much more clearer now! $\endgroup$ – McGuire Oct 25 '15 at 21:17
  • $\begingroup$ So does your answer totally eligible and I can say that it proves that there is no sample space which would consist only from an empty set? $\endgroup$ – ohidano Aug 9 '17 at 21:20
  • $\begingroup$ Sorry, I don't understand the question...what I am saying is that there exists no probability space whose sample space is the empty set. $\endgroup$ – Eric Wofsey Aug 11 '17 at 2:48

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