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?


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 $1$).

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.