Chapter 11

Abstract Measure And Integration11.1 σ Algebras

This chapter is on the basics of measure theory and integration. A measure is a real valuedmapping from some subset of the power set of a given set which has values in [0,∞]. Manyapparently different things can be considered as measures and also there is an integraldefined. By discussing this in terms of axioms and in a very abstract setting, many differenttopics can be considered in terms of one general theory. For example, it will turn out thatsums are included as an integral of this sort. So is the usual integral as well as things whichare often thought of as being in between sums and integrals.

Let Ω be a set and let F be a collection of subsets of Ω satisfying

/0 ∈F , Ω ∈F , (11.1.1)

E ∈F implies EC ≡Ω\E ∈F ,

If {En}∞n=1 ⊆F , then ∪∞

n=1 En ∈F . (11.1.2)

Definition 11.1.1 A collection of subsets of a set, Ω, satisfying Formulas 11.1.1-11.1.2 iscalled a σ algebra.

As an example, let Ω be any set and let F = P(Ω), the set of all subsets of Ω (powerset). This obviously satisfies Formulas 11.1.1-11.1.2.

Lemma 11.1.2 Let C be a set whose elements are σ algebras of subsets of Ω. Then ∩C isa σ algebra also.

Be sure to verify this lemma. It follows immediately from the above definitions but itis important for you to check the details.

Example 11.1.3 Let τ denote the collection of all open sets in Rnand let σ (τ) ≡ inter-section of all σ algebras that contain τ . σ (τ) is called the σ algebra of Borel sets . Ingeneral, for a collection of sets, Σ, σ (Σ) is the smallest σ algebra which contains Σ.

This is a very important σ algebra and it will be referred to frequently as the Borel sets.Attempts to describe a typical Borel set are more trouble than they are worth and it is noteasy to do so. Rather, one uses the definition just given in the example. Note, however, thatall countable intersections of open sets and countable unions of closed sets are Borel sets.Such sets are called Gδ and Fσ respectively.

Definition 11.1.4 Let F be a σ algebra of sets of Ω and let µ : F → [0,∞]. µ is called ameasure if

µ(∞⋃

i=1

Ei) =∞

∑i=1

µ(Ei) (11.1.3)

whenever the Ei are disjoint sets of F . The triple, (Ω,F ,µ) is called a measure spaceand the elements of F are called the measurable sets. (Ω,F ,µ) is a finite measure spacewhen µ (Ω)< ∞.

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