Today’s lecture came from Sections 1.7 and 2.1 in the book. We were introduced perhaps the most common -field associated with a given sample space , namely the Borel -field, which is the -field generated by the open sets of .
We did not discuss Section 1.8, but it is worth reading.
Moving on to Section 2.1, with the background in -fields now established, we are able to define probability measures, which are simply functions taking a -field of subsets of some outcome space to the interval . In order to be a probability measure, this function must satisfy three conditions: positivity, -additivity and the property that . We then establish several natural properties of probability measures.