 Probability space

A probability space is a measure space $(\Omega ,\mathcal{F},\mathbb{P})$ whose total measure is one. The sample space $\Omega$ contains the elementary outcomes; the sigma-algebra $\mathcal{F}$ contains the events to which the model assigns probabilities; the probability measure $\mathbb{P}$ supplies those values. The sigma-algebra lets us combine observable questions using complements and countable unions while remaining within the model.

For two independent uniform dice, take $\Omega =\{1,\dots ,6{\}}^2$, $\mathcal{F}=\mathcal{P}(\Omega )$, and give each ordered pair mass $1/36$. A random variable might record the sum, the larger result, or whether both results are even integers. These are different functions on the same sample space, rather than different elementary experiments. A probability distribution records the mass of the values of one such random variable and can forget information retained by the full probability space.