# Probability over a given $\sigma$-algebra

My question is very basic in a sens :

Given a set $\Omega$ and a $\sigma$-algebra $\mathscr T$ over $\Omega$, is it always possible to define a probability over $(\Omega, \mathscr T)$ ?

I assume my question is a little bit vague. Any piece of advise to precise it will be welcomed.

Take $x\in\Omega$, and put $P(A)=1$ if $x\in A$, $0$ otherwise. – Davide Giraudo Feb 5 '12 at 11:19
Assuming your set $\Omega$ is non-empty, you can always pick some $x\in\Omega$ and assign probability $1$ to those sets of $\mathscr T$ that contain $x$ and probability $0$ to those that don't. Maybe you did not mean this type of solution, but then you should be more specific about the conditions you want $\mathscr T$ to satisfy.