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Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces . Now, we will work in a more abstract and general setting, in which the Euclidean space is replaced by a more general space .
It turns out that in order to properly define measure and integration on a general space , it is not enough to just specify the set . One also needs to specify two additional pieces of data:
- A collection of subsets of that one is allowed to measure; and
- The measure one assigns to each measurable set .
For instance, Lebesgue measure theory covers the case when is a Euclidean space , is the collection of all Lebesgue measurable subsets of , and is the Lebesgue measure of .
The collection has to obey a number of axioms (e.g. being closed with respect to countable unions) that make it a -algebra, which is a stronger variant of the more well-known concept of a boolean algebra. Similarly, the measure has to obey a number of axioms (most notably, a countable additivity axiom) in order to obtain a measure and integration theory comparable to the Lebesgue theory on Euclidean spaces. When all these axioms are satisfied, the triple is known as a measure space. These play much the same role in abstract measure theory that metric spaces or topological spaces play in abstract point-set topology, or that vector spaces play in abstract linear algebra.
On any measure space, one can set up the unsigned and absolutely convergent integrals in almost exactly the same way as was done in the previous notes for the Lebesgue integral on Euclidean spaces, although the approximation theorems are largely unavailable at this level of generality due to the lack of such concepts as “elementary set” or “continuous function” for an abstract measure space. On the other hand, one does have the fundamental convergence theorems for the subject, namely Fatou’s lemma, the monotone convergence theorem and the dominated convergence theorem, and we present these results here.
One question that will not be addressed much in this current set of notes is how one actually constructs interesting examples of measures. We will discuss this issue more in later notes (although one of the most powerful tools for such constructions, namely the Riesz representation theorem, will not be covered until 245B).
A (concrete) Boolean algebra is a pair , where X is a set, and is a collection of subsets of X which contain the empty set , and which is closed under unions , intersections , and complements . The subset relation also gives a relation on . Because the is concretely represented as subsets of a space X, these relations automatically obey various axioms, in particular, for any , we have:
- is a partial ordering on , and A and B have join and meet .
- We have the distributive laws and .
- is the minimal element of the partial ordering , and is the maximal element.
- and .
We can then define an abstract Boolean algebra to be an abstract set with the specified objects, operations, and relations that obey the axioms 1-4. [Of course, some of these operations are redundant; for instance, intersection can be defined in terms of complement and union by de Morgan's laws. In the literature, different authors select different initial operations and axioms when defining an abstract Boolean algebra, but they are all easily seen to be equivalent to each other. To emphasise the abstract nature of these algebras, the symbols are often replaced with other symbols such as .]
Clearly, every concrete Boolean algebra is an abstract Boolean algebra. In the converse direction, we have Stone’s representation theorem (see below), which asserts (among other things) that every abstract Boolean algebra is isomorphic to a concrete one (and even constructs this concrete representation of the abstract Boolean algebra canonically). So, up to (abstract) isomorphism, there is really no difference between a concrete Boolean algebra and an abstract one.
Now let us turn from Boolean algebras to -algebras.
A concrete -algebra (also known as a measurable space) is a pair , where X is a set, and is a collection of subsets of X which contains and are closed under countable unions, countable intersections, and complements; thus every concrete -algebra is a concrete Boolean algebra, but not conversely. As before, concrete -algebras come equipped with the structures which obey axioms 1-4, but they also come with the operations of countable union and countable intersection , which obey an additional axiom:
As with Boolean algebras, one can now define an abstract -algebra to be a set with the indicated objects, operations, and relations, which obeys axioms 1-5. Again, every concrete -algebra is an abstract one; but is it still true that every abstract -algebra is representable as a concrete one?
The answer turns out to be no, but the obstruction can be described precisely (namely, one needs to quotient out an ideal of “null sets” from the concrete -algebra), and there is a satisfactory representation theorem, namely the Loomis-Sikorski representation theorem (see below). As a corollary of this representation theorem, one can also represent abstract measure spaces (also known as measure algebras) by concrete measure spaces, , after quotienting out by null sets.
In the rest of this post, I will state and prove these representation theorems. They are not actually used directly in the rest of the course (and they will also require some results that we haven’t proven yet, most notably Tychonoff’s theorem), and so these notes are optional reading; but these theorems do help explain why it is “safe” to focus attention primarily on concrete -algebras and measure spaces when doing measure theory, since the abstract analogues of these mathematical concepts are largely equivalent to their concrete counterparts. (The situation is quite different for non-commutative measure theories, such as quantum probability, in which there is basically no good representation theorem available to equate the abstract with the classically concrete, but I will not discuss these theories here.)