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Math 641 Lecture #19 ¶6.1,6.2,6.6

Definitions (6.1). Let M be a σ-algebra in a X.

A partition of E ∈ M is a countable collection {Ei} ⊂ M such that Ei ∩ Ej = ∅ for all ∞ i 6= j and E = ∪i=1Ei. A complex on M is a µ : M → C such that ∞ X µ(E) = µ(Ei) for all partitions {Ei} of E. (1) i=1 [NOTE: this condition is “countable additivity.”] A real (or signed) measure on M is a function µ : M → R which satisfies (1). Elementary Properties. (a) The class of real measures on M is a subclass of the complex measures on M. P (b) Since µ(E) ∈ C, the series i µ(Ei) converges for every partition {Ei} of E.

(c) Since relabeling the indices of the Ei gives another partition of E, the series P∞ i=1 µ(Ei) converges to the same quantity, µ(E), for every rearrangement of the µ(Ei)’s; hence the series is absolutely convergent. Examples. 1 R (a) If µ is a positive measure and f ∈ L (µ), then ν(E) = E f dµ is a . Pn Proof. Let {Ei} be any partition of E, and set fn = i=1 χEi f. 1 Then fn → f pointwise, and |fn| ≤ |f| ∈ L (µ). By LDCT, Z Z ν(E) = f dµ = lim fn dµ E n→∞ E n n Z X X Z = lim χE f dµ = lim χE f dµ n→∞ i n→∞ i E i=1 i=1 E n n X Z X = lim f dµ = lim ν(Ei) n→∞ n→∞ i=1 Ei i=1 ∞ X = ν(Ei). i=1

1 R (b) If µ is a positive measure and f ∈ L (µ) is extended real-valued, then ν(E) = E f dµ is a real measure. [Same argument as in (a) applies here.] Problem. If µ is a complex measure on M, is there a positive measure λ on M such that |µ(E)| ≤ λ(E) for all E ∈ M? If there is such a positive measure, it must satisfy

∞ ∞ X X λ(E) = λ(Ei) ≥ |µ(Ei)| i=1 i=1 for every partition {Ei} of any E ∈ M. [NOTE: the equality is countable additivity of a positive measure on countable collections of pairwise disjoint measurable sets; the inequality is λ(E) ≥ |µ(E)| for ALL E ∈ M; P∞ the last series converges since i=1 µ(Ei) converges absolutely.] A candidate for λ is the

∞  X  |µ|(E) = sup |µ(Ei)| : {Ei} is a partition of E . i=1

Certainly |µ|(E) ≥ |µ(E)| [take E1 = E, Ei = ∅ for i > 1], but equality need not hold. However, if |µ| is a positive measure, then it is the smallest of all positive measures λ that satisfy |µ(E)| ≤ λ(E) for all E ∈ M [this follows from the supremum in the definition of |µ|].

Theorem (6.2). If µ is a complex measure on M, then |µ| is a positive measure on M.

Proof. For a partition {Ei} of E ∈ M, let ti ∈ R satisfy ti < |µ|(Ei).

By the supremum in the definition of |µ| there is for each i a partition {Aij} of Ei such that X ti < |µ(Aij)| ≤ |µ|(Ei). j

The countable collection {Aij} is a partition of E, so that X X ti ≤ |µ(Aij)| ≤ |µ|(E). i i,j

It follows that X  X  |µ|(Ei) = sup ti : ti < |µ|(Ei) ≤ |µ|(E). i i

To prove the opposite inequality, let {Ei} and {Aj} be partitions of E.

For any fixed j, the collection {Aj ∩ Ei} is a partition of Aj, and for any fixed i, the collection {Aj ∩ Ei} is a partition of Ei. Then

X X X |µ(Aj)| = µ(Aj ∩ Ei) [use partition of Aj]

j j i X X ≤ |µ(Aj ∩ Ei)| j i X X ≤ |µ(Aj ∩ Ei)| [switch order of summation] i j X ≤ |µ|(Ei) [definition of |µ|(Ei)]. i It follows that  X  X |µ|(E) = sup |µ(Aj)| : {Aj} a partition of E ≤ |µ|(Ei). j i

Thus, |µ| is countably additive. The set function |µ| is not identically ∞ because, using the only partition of ∅, X µ(∅) = µ(∅) and |µ(∅)| < ∞ ⇒ µ(∅) = 0 ⇒ |µ|(∅) = 0. i

Therefore, |µ| is a positive measure on M.  Definition. The positive measure |µ| is the total variation of a complex measure µ.

Observation. If µ is a positive measure, then the set function |µ| is equal to µ by countable additivity; hence the “total variation” of µ is itself. Definitions (6.6) Let µ be a real measure. The positive and negative variations of µ are the positive measures

µ+ = (1/2)|µ| + µ and µ− = (1/2)|µ| − µ.

Clearly, |µ| = µ+ + µ−. The Jordon decomposition of µ is µ = µ+ − µ−. Definitions Let X be a LCH space.

A complex (real) Borel measure is a complex (real) measure defined on BX . A complex (real) Borel measure µ is regular if |µ| is a regular positive Borel measure on BX .