Measure Theoretic Probability - Video 7 - Algebras
Algebras
An algebra is any sub-collection \(\mathcal A\subseteq \mathcal P(\Omega)\) which
contains the universe, \(\Omega\in\mathcal A\)
is closed under complements, \(E\in\mathcal A\Rightarrow E^c\in\mathcal A\)
is closed under pairwise unions, \(E_1,E_2\in\mathcal A\Rightarrow E_1\cup E_2 \in\mathcal A\).
This bundle of properties is something that any collection of measurables should have. You should be able to measure the universe. In the case of length measure it should be \(\infty\), and in the case of any probability measure it should be 1. Regardless of the specific measure, though, the entire universe should be one of the measurables.
If you can measure a set, you should be able to measure its complement. If you can measure two sets, you should be able to measure their union. So the idea of an algebra encodes what we expect and need from any collection of measurables.
Why These Properties?
If you’re wondering why we choose these properties (i.e. why not include more properties, or why not generalize further by throwing some out), I can’t answer that with perfect certainty, but I have what I think is a plausible story:
Historically, measure theory began with Lebesgue’s theory of length measure. After Vitali proved the impossibility of a measure of all real functions, Lebesgue devised a definition of the measurable subsets of real numbers, and proceeded to prove all sorts of theorems about them and the length-measure function.
Mathematicians then wanted to generalize not just the idea of length-measure, but also the proofs that we successfully made about length-measure. It was observed that these closure properties, as well as containing the universe, came up in those proofs in an indispensable way. Therefore, mathematicians felt it natural and in a sense necessary to require these properties, so that the generalization could be productive.