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Thus far, we have only focused on measure and integration theory in the context of Euclidean spaces {{\bf R}^d}. Now, we will work in a more abstract and general setting, in which the Euclidean space {{\bf R}^d} is replaced by a more general space {X}.

It turns out that in order to properly define measure and integration on a general space {X}, it is not enough to just specify the set {X}. One also needs to specify two additional pieces of data:

  1. A collection {{\mathcal B}} of subsets of {X} that one is allowed to measure; and
  2. The measure {\mu(E) \in [0,+\infty]} one assigns to each measurable set {E \in {\mathcal B}}.

For instance, Lebesgue measure theory covers the case when {X} is a Euclidean space {{\bf R}^d}, {{\mathcal B}} is the collection {{\mathcal B} = {\mathcal L}[{\bf R}^d]} of all Lebesgue measurable subsets of {{\bf R}^d}, and {\mu(E)} is the Lebesgue measure {\mu(E)=m(E)} of {E}.

The collection {{\mathcal B}} has to obey a number of axioms (e.g. being closed with respect to countable unions) that make it a {\sigma}-algebra, which is a stronger variant of the more well-known concept of a boolean algebra. Similarly, the measure {\mu} 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 {(X, {\mathcal B}, \mu)} 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).

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