# Tag Archives: sigma-field

## The Borel sigma-field; Probability measures

Today’s lecture came from Sections 1.7 and 2.1 in the book.  We were introduced perhaps the most common $\sigma$-field associated with a given sample space $\Omega$, namely the Borel $\sigma$-field, which is the $\sigma$-field generated by the open sets of $\Omega$.

We did not discuss Section 1.8, but it is worth reading.

Moving on to Section 2.1, with the background in $\sigma$-fields now established, we are able to define probability measures, which are simply functions taking a $\sigma$-field of subsets of some outcome space $\Omega$ to the interval $[0,1]$.  In order to be a probability measure, this function must satisfy three conditions: positivity, $\sigma$-additivity and the property that $\mathbb{P}(\Omega) = 1$.  We then establish several natural properties of probability measures.

## The sigma-field generated by a given class of observations

Today we looked section 1.6 of the book.  In the first lecture, we came to understand $\sigma$-fields as being the fundamental mathematical structure of collections of inferences based on some set of observations.  We call the set of all possible inferences possible from a class of observations $\mathcal{C}$ the $\sigma$-field generated by $\mathcal{C}$, denoted $\sigma(\mathcal{C})$.

In the examples we looked at, the sample space (and the collection of observations) were finite.  A $\sigma$-field generated by a finite class can always be constructed by enumeration.  However, when the class is infinite, this may not be possible.  The major result in Section 1.6 is to show that such a $\sigma$-field always exists and can be constructed as an intersection of larger sets.

Topic 1a: Sigma-fields