一題外測度證明 (計數測度是外測度)

Question

Define μ(E) as the number of points in E if E is finite and μ(E)=∞ if E is infinite. Show that μ is an outer measure. Determine the measurable sets.

註：

An extended real-valued set function v is said to be \"countably subadditive\" if

v(∪j=1~∞ E_j) <= Σj=1~∞ v(E_j)

whenever the E_j and their union belong to the domain of v.

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An extended real-valued set funciton v is said to be \"monotone\" if v(E)<=v(F) whenever E, F are in the domain of v and E包含於F.

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An \"outer measure\" is an extended real-valued set function μ having the following properties:

(i) The domain of μ consists of all the subsets of X (universal set).

(ii) μ is nonnegative.

(iii) μ is countably subadditive.

(iv) μ is monotone.

(v) μ(空集合)=0.

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Given an outer measure μ, we say that a set E is \"μ-measurable\" (or, briefly, \"measurable\") if

μ(A)=μ(A∩E)+μ(A-E)

for any subset A of X

Answer 1

最佳解答

In proving the following two claims, I will use the fact that counting measure is finitely additive for disjoint finite sets.Claim.  μ is an outer measure on X.Proof.  Let's verify the five conditions.(i) Domain.μ is defined on the domain of the collection of subsets of X.(ii) Nonnegativity.μ is clearly nonnegative.(iii) Countable subadditivity.Let En be a countable collection of subsets of X. There are two cases:If ∑n μ(En) = ∞, then μ(∪n En) ≤ ∞ = ∑n μ(En).Otherwise, ∑n μ(En) < ∞, so En must be finite for every n. Furthermore, μ(En) must be a nonnegative integer for every n, so there exists N ∈ N such that En = ∅ and μ(En) = 0 for all n > N.μ(∪n En) = μ(∪n≤N En) = ∑n≤N (μ(En) - μ((∪j≤n Ej)∖En))≤ ∑n≤N μ(En) = ∑n μ(En).(iv) Monotonicity.If E and F are subsets of X and E ⊆ F, there are two cases:If F is infinite, then μ(E) ≤ ∞ = μ(F).Otherwise, F is finite, and so is E. E has less elements than F, so μ(E) ≤ μ(F).(v) The empty set.

μ(∅) = 0 since ∅ is a finite set of 0 elements.We conclude that μ is an outer measure on X. ∎The measurable sets: which subsets E ⊆ X satisfyμ(A) = μ(A∩E) + μ(A∖E)     (*)for all A ⊆ X?Claim.  All subsets E of X are measurable.Proof.  Let E and A be subsets of X.First, note that A∩E and A∖E are disjoint and A = (A∩E)∪(A∖E).By subadditivity, we haveμ(A) ≤ μ(A∩E) + μ(A∖E).Hence, it is sufficient to prove μ(A) ≥ μ(A∩E) + μ(A∖E).If A is infinite, then μ(A) = ∞, and so this inequality is already satisfied.Otherwise, A is finite, and so A∩E and A∖E are both finite. Therefore, μ(A) = μ(A∩E) + μ(A∖E).Since (*) has been showed for arbitrary subsets E and A of X, we conclude that all subsets of X are measurable. ∎