My first post in this series about Lebesgue integration presents the concepts of -algebras and measurability along with some basic results along these lines.

The parallel Rudin draws between -algebras and topologies is a pretty nice one. Whereas the sets in a topology are closed under arbitrary union and finite intersection, those in a -algebra are closed under complements and countable union (and by implication, countable intersection and subtraction).

Formally, an **algebra** on a set is defined to be a collection of subsets of such that i) , ii) , iii) .

A **-algebra** shares properties i) and ii), but . This condition of closure under not just finite, but countable union, gives -algebras relevance in the study of integration.

In fact, it’s not terribly uncommon to arrive upon a nontrivial -algebra given an arbitrary collection of subsets of :

Result 1: If is a collection of subsets of , there exists a unique smallest -algebra containing this collection. This -algebra is said to be *generated*** **by .

Proof: Just take the intersection of all -algebras that contain (at the very least, the collection of all subsets of is in this family).

Any space where we can define such a structure is called a **measurable space**; it is easy to see that is a measurable space when endowed with the -algebra consisting of the unions of intervals. The sets in such a collection are the **measurable sets, **and functions from a measurable space to a topological space are **measurable functions** if they pull the open sets of back to measurable sets of . Quickly convince yourself that many familiar Riemann-integrable functions are measurable.

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