12.14. EXERCISES 337

2. Verify that µ defined in Lemma 12.1.9 is an outer measure.

3. Let F : R→ R be increasing and right continuous. Let Λ f ≡∫

f dF where the in-tegral is the Riemann Stieltjes integral of f . Show the measure µ from the Rieszrepresentation theorem satisfies

µ ([a,b]) = F (b)−F (a−) ,µ ((a,b]) = F (b)−F (a) ,

µ ([a,a]) = F (a)−F (a−) .

4. Let Ω be a metric space with the closed balls compact and suppose µ is a measuredefined on the Borel sets of Ω which is finite on compact sets. Show there exists aunique Radon measure, µ which equals µ on the Borel sets.

5. ↑ Random vectors are measurable functions X, which map a probability space to Rn.A probability space is of the form (Ω,P,F ). Thus X(ω) ∈ Rn for each ω ∈ Ω andP is a probability measure defined on the sets of F , a σ algebra of subsets of Ω. ForE a Borel set in Rn, define

µ (E)≡ P(X−1 (E)

)≡ probability that X ∈ E.

Show this is a well defined measure on the Borel sets of Rn and use Problem 4 toobtain a Radon measure, λ X defined on a σ algebra of sets of Rn including the Borelsets such that for E a Borel set, λ X (E) =Probability that (X ∈E).

6. Suppose X and Y are metric spaces having compact closed balls. Show

(X×Y,dX×Y )

is also a metric space which has the closures of balls compact. Here

dX×Y ((x1,y1) ,(x2,y2))≡max(d (x1,x2) ,d (y1,y2)) .

LetA ≡ {E×F : E is a Borel set in X ,F is a Borel set in Y} .

Show σ (A ), the smallest σ algebra containing A contains the Borel sets. Hint:Show every open set in a metric space which has closed balls compact can be ob-tained as a countable union of compact sets. Next show this implies every open setcan be obtained as a countable union of open sets of the form U×V where U is openin X and V is open in Y .

7. Suppose (Ω,S ,µ) is a measure space which may not be complete. Could you obtaina complete measure space,

(Ω,S ,µ1

)by simply letting S consist of all sets of the

form E where there exists F ∈S such that (F \E)∪ (E \F) ⊆ N for some N ∈Swhich has measure zero and then let µ (E) = µ1 (F)?

8. If µ and ν are Radon measures defined onRn andRm respectively, show µ×ν is alsoa radon measure on Rn+m. Hint: Show the µ×ν measurable sets include the opensets using the observation that every open set in Rn+m is the countable union of sets