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

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