Probability space concepts

A field is a non-empty class of subsets of $\Omega$ closed under finite union, finite intersection and complements.  A synonym for field is algebra.

From de Morgan's laws a field is also closed under finite intersection.

A $\sigma$-field $\mathcal{B}$ is a non-empty class of subsets of $\Omega$ closed under countable union, countable intersection and complements.  A synonym for $\sigma$-field is $\sigma$-algebra.

In probability theory, the event space is a $\sigma$-field.  This allows us enough flexibility constructing new-events from old ones (closure) but not so much flexibility that we have trouble assigning probabilities to the elements of the $\sigma$-field.

For the Reals, we start with sets that we know how to assign probabilities.

Supposes $\Omega=\mathbb{R}$ and let

$\mathcal{C}=\{(a,b],-\infty \leq a \leq b < \infty \}$

The Borel sets is defined as

$\mathcal{B}(\mathbb{R}) \equiv \sigma(\mathcal{C})$

Also one can show that

$\mathcal{B}(\mathbb{R}) = \sigma(\text{open sets in } \mathbb{R})$