# Can a sample space consist of only the empty set?

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 $1$).