While I was in Indiana a few weeks ago at Notre Dame for their *Geometry & Topology* conference I had a very enlightening conversation with my roommate at the time (and now good friend). He had come across a small section in one of Rudin’s analysis textbooks which highlighted an analogy between measurable spaces and topological spaces. I’d like to dive into that analogy in what follows.

#### Measurable Spaces & Topological Spaces

The analogy begins right with the definition of a measurable space and a topological space. So, lets restate these definitions. Given a set , a set of subsets of , denoted , is called a **-algebra** on if the following three properties are satisfied:

1. .

2. If then .

3. If for , then .

In other words, * is nonempty, closed under complements, and closed under countable unions*. It’s easy to check , and that is closed under finite unions, countable intersections, and finite intersections. We call the pair a **measurable space**, but we often refer to as a measurable space without mention of the -algebra.

Given a set , a set of subsets of , denoted , is called a **topology** on if the following three properties are satisfied:

1. .

2. If for , then .

3. If for , then .

In other words, * is nonempty, closed under arbitrary unions, and closed under finite intersections*. We call the sets the **open sets** of , and we call a subset of **closed** if its complement is open. The pair is called a **topological space** and we often refer to as a topological space without mentioning the topology explicitly.

The similarity of these two concepts already starts to shine as a -algebra and a topology are both a collections of subsets of the ambient space that are nonempty and satisfy two closure properties. One might then ask the question if there exists a space that is both a measurable space and a topological space such that . This is true as we will see.

#### Maps on Measurable Spaces and Topological Spaces

Mathematical spaces are important in their own right, but we often learn much more information about a space by studying the structure-preserving maps to-and-from it. For measurable spaces our maps are the measurable functions, and for topological spaces they are the continuous functions.

We say that a map between two measurable spaces and is **measurable** if the preimage is in the -algebra for for every set in the -algebra for . Similarly, a map between two topological spaces is **continuous** if is an open set in for every open set in .

The reason these definitions are so similar is that the structure of measurable spaces and topological spaces is entirely dependent on a collection of subsets of the underlying space and inverse images respect arbitrary unions, arbitrary intersections, and complements.

#### Measurable Spaces Generated by A Topology

Here’s an easy fact: If is a set, any collections of its subsets gives rise to a -algebra on such that the collection of subsets is contained in the -algebra. We call this -algebra the -algebra **generated by **. The idea is that you take the intersection of all the -algebras on containing (this intersection is nonempty because the set of all subsets of is a -algebra) and prove that this is a -algebra.

Something especially interesting happens if we let be a topological space. Indeed, if is a topological space then the topology of gives rise to a -algebra on so that we may treat as a measurable space as well. In this case, every open and closed set belongs to and we call the members of the **Borel sets** of . If is a topological space then we can treat it as a measurable space with the Borel sets being the measurable sets. In this case we call a **Borel space**. If and are two Borel spaces then any continuous function is measurable since preimages respect arbitrary unions, arbitrary intersections, and complements.

#### Measures

Before we look at a couple of examples, there’s another important topic that needs some discussion, namely positive measures. Given a measurable space a **positive measure** (or simply a measure) on is a function that is **countably additive**. That is, if is a collection of pairwise disjoint measurable sets, then

.

It is usually required to assume to weed-out odd examples. It’s easy to show that the measure satisfies some basic properties such as (measurable sets with we call **sets of measure **) and that if and are measurable sets with then . Intuitively, *we can think of a measure as a function which assigns a “size” to the measurable sets* on the ambient space. If we have a measurable space with a measure , then is called a measure space with measure .

#### Examples

1. Let be any set and let be the power set of . Then is a measurable space. If , define where if is infinite. is called the counting measure on , making into a measure space.

2. Let and again let be the power set of . We can make into a measure space by giving it the measure . Since is countably additive . This is a prototypical example of a probability space (a measure space where ), and its the space used to model a fair coin flip since either the coin is heads (0) or tails (1) and each event has an equal chance of occurring ().

3. Let . Then there exists a measure called the **Lebesgue measure** which is translation invariant, every subset of a set of measure is measurable and has measure (a measure with this property is called complete). In essence, this measure encompasses the idea of measuring -dimensional volume to a measurable subset of .

4. Let be a locally compact Hausdorff topological group. Then there exists a measure called the **Haar measure** which we can think of as assigning “volume” to Borel subsets of such that is translation invariant and complete. Since with the usual topology is locally compact Hausdorff it has a Haar measure, and this Haar measure is precisely the Lebesgue measure.

I hope you now enjoy the elegance of the analogy between measurable spaces and topological spaces as much as I do! That’s all for now.

#### References

*Papa Rudin *– Walter Rudin