MATH & MATH FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING Scientia Imperii Decus et Tutamen 1

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1 MATH & MATH FUNCTIONS OF A REAL VARIABLE EXERCISES FALL 2015 & SPRING 2016 Scientia Imperii Decus et Tutamen 1 Robert R. Kallman University of North Texas Department of Mathematics 1155 Union Circle Denton, Texas office telephone: ; office fax: ; office kallman@unt.edu August 19, Taken from the coat of arms of Imperial College London.

2 Fall 2015 Exercise 1. Recall that if X is a set and A and B are subsets of X, then A B = (A \ B) (B \ A). Prove that χ A B = χ A χ B c + χ A cχ B. Use characteristic functions to prove that A = A, A B = B A, A (B C) = (A B) (A C), A c B c = A B, (A B) C = A (B C), and that the set of subsets X forms a commutative ring with identity with * = and + =. Hint: prove that χ (A B) c = χ A χ B + χ A cχ B c. Exercise 2. If X is a set and A is an algebra of subsets of X, let S be the smallest colection of subsets of X that contains A and is closed under complements and unions of increasing sequences. Then S = m X (A) = σ X (A). Exercise 3. Problem # 4, page 7 of Cohn. Show that if A is an algebra of sets, and if n 1 A n belongs to A whenever {A n } n 1 is a sequence of disjoint sets in A, then A is a σ-algebra. Exercise 4. Problem # 7, page 7 of Cohn. Let S be a nonempty collection of subsets of the set X. Show that for each A in σ(s) there is a countable subfamily S 0 of S such that A σ(s 0 ). Exercise 5. Problem # 8, page 14, of Cohn. Let (X, A, µ) be a measure space and let µ : A [0, + ] by µ (A) = lub{µ(b) B A, B A, and µ(b) < + }. Show that µ is a measure on (X, A). Show that if A A and A is σ-finite under µ, then µ (A) = µ(a). In particular, if µ is σ-finite, then µ = µ. Find µ if X is nonempty and µ is the measure defined by µ(a) = + if A A and A, µ( ) = 0. Exercise 6. Problem # 9, page 14, of Cohn. Let (X, A, µ) be a measure space and let {A k } k 1 A be a sequence such that k 1 µ(a k) < +. Show that the set of points that belong to A k for infinitely many values of k has µ- measure zero. 1

3 Exercise 7. Let X be a set, B a collection of subsets of X such that, X B, ψ : B [0, + ] a function such that ψ( ) = 0. If S P(X), define ψ (S) = glb{ n 1 ψ(b n) B n B for n 1 and S n 1 B n }. Prove that ψ is an outer measure on X. Hints: Note that the glb exists and makes sense because, X B. Note further that we could start with any collection B of subsets of X and any function ψ : B [0, + ], for we can always toss and/or X into B and extend ψ by setting ψ( ) = 0 and/or ψ(x) = +. Exercise 8. Let {µ γ} γ Γ be a nonempty collection of outer measures on the set X. Then µ (A) = lub γ Γ µ γ(a) is an outer measure on X. In particular the maximum of two outer measures on X is an outer measure on X. Exercise 9. Let {µ γ} γ Γ be a nonempty collection of outer measures on the set X, let a γ [0, + ] (γ Γ) and define µ (A) = γ Γ a γµ γ(a) for A X. Prove that µ is an outer measure on X. Exercise 10. Consider the Prime Example??. A subset B X is µ -measurable if and only if µ (A) = µ (A B) + µ (A B c ) for every A A. Exercise 11. Let X be a set, µ an outer measure on X, and B X. Prove that B is µ -measurable if and only if µ (A 1 A 2 ) = µ (A 1 ) + µ (A 2 ) for all subsets A 1, A 2 X with A 1 B and A 2 B =. Hint: the proof is very easy. Exercise 12. Apply the construction of Exercise 7 to the special case in which X is R, B is the collection of bounded open subintervals of R, and ψ( a, b ) = b a. Let ψ be the corresponding outer measure. Show that ψ is Lebesgue outer measure. More generally, if X is an open subinterval of R, φ is a nondecreasing continuous function on X, µ φ is the corresponding countably additive measure on A, the algebra generated by the subintervals of X, B is the set of all open subintervals of X whose closure is contained in X, and ψ φ : B [0, + ] is defined by ψ φ ( a, b ) = φ(b) φ(a), then µ φ = ψ φ. Exercise 13. Let (X, d) be a metric space and let µ be an outer measure on X. Show that if every Borel subset of X is µ -measurable, then µ is a metric outer measure. Hint: let C = cl X (A 1 ) and use Exercise 11. 2

4 Exercise 14. Let (X, d) be a metric space, α 0 and ϵ > 0 be positive numbers. Perform the construction of Exercise 7 to X, B = {A A X, diam(a) < ϵ}, and ψ(a) = diam(a) α (for convenience we set diam( ) 0 = 0 and diam(a) 0 = 1 if A ) for each A B to obtain an outer measure denoted by H α (ϵ) on X. Show that if ϵ ϵ, then H α (ϵ) (A) H (ϵ ) α (A) holds for every subset A X. Show that if 0 α β and 0 < ϵ 1, then H (ϵ) (A) H(ϵ) α (A) for all A X. β Exercise 15. Let X be a set, Γ a directed set and {µ γ} γ Γ a net of outer measures on X such that γ 1 γ 2 = µ γ 1 (A) µ γ 2 (A) for each A X. Then µ (A) = lim γ Γ µ γ(a) is an outer measure on X. Exercise 16. If (X, d) is a metric space and α 0, prove that H α (A) = lim ϵ 0 H α (ϵ) (A) is a metric outer measure on X. H α is called the α-dimensional Hausdorff outer measure on X and the restriction of H α to the σ-algebra of the H α -measurable subsets of X is called the α-dimensional Hausdorff measure on X. Show that if 0 α β, then H β (A) H α (A) for every A X. Exercise 17. Let φ be a nondecreasing (not necessarily continuous) real valued function on an interval J R. Apply the construction of Exercise 7 to X = J, B = { a, b] a, b J, a < b}, and ψ φ ( a, b]) = φ(b) φ(a). If J has a left hand endpoint a, then we also include {a} in B and set ψ φ ({a}) = 0. Prove that ψ φ is a metric outer measure on J. ψ φ is called the Lebesgue-Stieltjes outer measure corresponding to φ, and its restriction to the σ-algebra of ψ φ-measurable sets is called the Lebesgue-Stieltjes measure corresponding to φ. If [a, b] J then ψ φ( a, b]) φ(b) φ(a). ψ φ({a}) = 0 if a is the left hand endpoint of J, if J indeed has a left hand endpoint. ψ φ({a}) = φ(a) φ(a ) if a J is not a left hand endpoint. ψ φ(k) < + if K J is compact. ψ φ has bounded range and ψ φ B(J) is a bounded Borel measure if φ is bounded. Exercise 18. Use the notation of Exercise 17. If φ is a nondecreasing function on J which is rightcontinuous, then ψφ( a, b]) = φ(b) φ(a) if a, b J and a < b. If φ is a continuous nondecreasing function on J, then ψφ(a) = µ φ(a) for every subset A J. In particular, if φ(x) = x and J = R, then ψφ is Lebesgue outer measure. Exercise 19. Let µ be a bounded Borel measure on R, and let φ(x) = µ(, x]) for x R. Then φ 3

5 is bounded, nondecreasing, right-continuous, φ( ) lim x φ(x) = 0, and µ({x}) = φ(x) φ(x ). φ is continuous if and only if µ is a continuous bounded Borel measure. Conversely, let φ be a bounded, nondecreasing, right-continuous function on R, normalized so that lim x φ(x) = 0. Then ψφ B(R) is a bounded Borel measure on R, and φ(x) = ψφ(, x]) for all x R. Let S be the set of bounded, nondecreasing, right-continuous functions on R, normalized so that lim x φ(x) = 0. Let T be the set of bounded Borel measures on R. Then the mapping φ ψφ B(R) is a bijection from S T. Exercise 20. Let φ be a bounded nondecreasing right-continuous function on R. Then φ may be uniquely decomposed as φ = φ c + φ d, where φ c is a bounded continuous nondecreasing function on R and where φ d has the following form: there exists an at most countable subset C R and a function f : C [0, + such that c C f(c) < + and φ d(x) = c (C,x]) f(c). Exercise 21. Let X be a topological space, A a σ-algebra on X such that A B(X), and µ a measure on A such that the triple (X, A, µ) is a regular measure space. Show that the completion (X, A µ, µ) of (X, A, µ) is also a regular measure space. Exercise 22. Prove that the supremum of an uncountable family of [, + ]-valued Borel functions can fail to be Lebesgue measurable. Exercise 23. Prove that if f : R R is everywhere differentiable on R, then f is a Borel function. Exercise 24. Let (X, A) be a measurable space and let {f n } n 1 be a sequence of [, + ]-valued measurable functions on X. Prove that {x X lim n + f n (x) exists and is finite} A. Exercise 25. If N R is a Lebesgue null set then N c is dense in R. Let f, g : R R be continuous and let µ be Lebesgue measure. Prove that if f = g µ-a.e. then f = g. Exercise 26. Let (X, A, µ) be a measure space, let A A, let µ A (B) = µ(a B) for all B A and let f : X [0, + ] be measurable. Then µ A is a countably additive measure on A and fdµa = fχ A dµ = A fdµ. 4

6 Supplementary Exercises Supplemental Exercise 1. Consider the construction of Exercise 7. Prove that if A B, then ψ (A) ψ(a). Prove that a subset B of X is ψ -measurable if and only if ψ (A) = ψ (A B) + ψ (A B c ) for every A B such that ψ (A) < +. Conclude that a subset B of R is Lebesgue measurable if and only if λ (I) = λ (I B) + λ (I B c ) for every bounded open subinterval I of R, where λ is Lebesgue outer measure. Supplemental Exercise 2. Let X be a set, µ an outer measure on X, A X, B a µ -measurable subset of X, and A B =. Prove that µ (A B) = µ (A) + µ (B). Supplemental Exercise 3. Let X be a set, µ an outer measure on X and Y a µ -measurable subset of X. Prove that a subset B of Y is µ -measurable if and only if µ (A) = µ (A B) + µ (A B c ) for every subset A of Y. Supplemental Exercise 4. Let X be a set, µ an outer measure on X, and Y a µ -measurable subset of X such that µ (Y ) < +. Let B be a subset of Y such that µ (Y ) = µ (B) + µ (Y B c ). Prove that µ (A) = µ (A B) + µ (A B c ) for every µ -measurable subset A of Y. Supplemental Exercise 5. Let X be a set, µ an outer measure on X, and Y a µ -measurable subset of X such that µ (Y ) < +. Let B be a subset of Y such that µ (Y ) = µ (B) + µ (Y B c ). Prove that µ (A) = µ (A B) + µ (A B c ) for every µ -measurable subset A of X. Supplemental Exercise 6. Consider the Prime Example??. Suppose that Y is a µ -measurable subset of X such that µ (Y ) < +. Prove that a subset B of Y is µ -measurable if and only if µ (Y ) = µ (B) + µ (Y B c ). Conclude that a subset B of a bounded subinterval I of R is Lebesgue measurable if and only if λ (I) = λ (B) + λ (I B c ), where λ is Lebesgue outer measure. Historical note: this shows that our notion of Lebesgue measurability coincides with Lebesgue s original definition. Supplemental Exercise 7. Let A be a subset of R such that µ (A) = 0, where µ is Lebesgue outer measure. Prove that if B = {x 2 x A}, then µ (B) = 0. 5

7 Supplemental Exercise 8. Let A be a Lebesgue measurable subset of R and let µ be Lebesgue measure. Then for every ϵ > 0, there is an open set U and a closed set F such that F A U, µ(u A) < ϵ and µ(a F ) < ϵ. Supplemental Exercise 9. Any subset of R which has positive outer Lebesgue measure contains a set which is not Lebesgue measurable. Supplemental Exercise 10. Let λ R, let A R, let µ be Lebesgue outer measure, and let λa = {λa a A}. Then µ (λa) = λ µ (A) (here, as is usual in measure theory, we interpret 0 + = 0). Show that if A is Lebesgue measurable, then λa is Lebesgue measurable, and if A is a Borel set, then λa is a Borel set. Supplemental Exercise 11. Let µ be Lebesgue measure on R and let A R be Lebesgue measurable with µ(a) < +. Let ϵ > 0. Then there a set B R which is a finite disjoint union of bounded intervals such that µ(a B) < ϵ. Supplemental Exercise 12. If (X, A, µ) is a measure space and if P, Q, R, and S A have finite µ-measure, prove that µ(p Q) µ(r S) µ(p R) + µ(q S). Supplemental Exercise 13. Let µ be Lebesgue measure on R, and let A and B be Lebesgue measurable subsets of R such that µ(a) < + and µ(b) < +. Prove that φ(x) = µ(a (B + x)), φ : R R, is a continuous function. Supplemental Exercise 14. Use Supplemental Exercise 13 to give another proof of Steinhaus theorem. Supplemental Exercise 15. Let µ be Lebesgue measure on R, let {x n } n 1 be an unbounded sequence of real numbers and let A be a Lebesgue measurable subset of R such that µ(a) < + and such that µ(a (A+x n )) = 0 for all n 1. Prove that µ(a) = 0. Supplemental Exercise 16. Let µ be a countably additive measure on B(R) which is finite on compact sets. Suppose that D is a dense subset of R and that µ(a + x) = µ(a) for every Borel set A and every x D. Prove that µ is the restriction of a nonnegative scalar multiple of Lebesgue measure to B(R). 6

8 Supplemental Exercise 17. Let µ be Lebesgue measure on R and let A be a Lebesgue measurable subset of R such that µ(a (A + x)) = 0 for a dense set of real numbers x. Prove that either µ(a) = 0 or µ(a c ) = 0. Supplemental Exercise 18. If X is a topological space, recall that a subset of X which is the intersection of a sequence of open subsets of X is called a G δ subset of X. Consider the construction of Exercise 7 and suppose that X is a topological space. If each element of B is an open subset of X, then for every subset A of X there is a G δ subset B of X such that A B and ψ (A) = ψ (B). Conclude in particular that each H α (ϵ) (α 0, ϵ > 0) has this property if X is a metric space. If each element of B is a Borel subset of X, then for every subset A of X there is a Borel subset B of X such that A B and ψ (A) = ψ (B). Supplemental Exercise 19. Let X be a topological space and let µ n (n 1) be a nondecreasing sequence of outer measures on X, and let µ (A) = lim n + µ n(a) for every subset A of X. We know from Exercise 15 that µ is an outer measure on X. Suppose each µ n (n 1) has the property that for every subset A of X there is a G δ subset B n of X such that A B n and µ n(a) = µ n(b n ). Prove that µ has the same property. Use Supplemental Exercise 18 to conclude that H α has this property. Suppose each µ n (n 1) has the property that for every subset A of X there is a Borel subset B n of X such that A B n and µ n(a) = µ (B n ). Prove that µ has the same property. Supplemental Exercise 20. Let (X, d) be a metric space, let H α be the α-dimensional Hausdorff outer measure on X, and let A X. Prove that if H α (A) < +, then H β (A) = 0 if β > α. If H α (A) > 0, then H β (A) = + for β < α. Supplemental Exercise 21. Use the notation of Supplemental Exercise 20. If A X, we define the Hausdorff dimension of A as follows: dim(a) = 0 if H α (A) = 0 for all α > 0; otherwise, let dim(a) = lub{α H α (A) = + } = glb{α H α (A) = 0} (these two agree by Supplemental Exercise 20). Prove that if A B, dim(a) dim(b). Prove that dim(r) = 1. Supplemental Exercise 22. Let (X, d) be a metric space, and let dim( ) : P(X) [0, + ] be the Hausdorff dimension function. Show that if A n X and α dim(a n ) β for n 1, then α dim( n 1 A n ) β. 7

9 Supplemental Exercise 23. Let J be a subinterval of R, let φ be a real valued nondecreasing function on J, let ψ φ be the corresponding Lebesgue-Stieltjes outer measure, M the σ-algebra of ψ φ-measurable subsets of J, and λ = ψ φ B(J) the restriction of ψ φ to the Borel subsets of J. Prove that the completion of (J, λ, B(J)) is (J, ψ φ, M). Supplemental Exercise 24. Let µ be a countably additive measure on B(R) which is finite on compact sets. Let A be a Borel set, and let ϵ > 0. Prove that there exists a closed set F and an open set U such that F A U and µ(u F ) < ϵ. In particular, every Lebesgue-Stieltjes measure on R has this property. 8

10 Spring 2016 Exercise 27. Let [a, b] R, let µ be Lebesgue measure on R and let f : [a, b] R be continuous on [a, b] and 0 on R [a, b]. Then f is integrable on R and fdµ = fdµ = b f(x)dx, the [a,b] a Riemann integral of f over [a, b]. Exercise 28. Let (X, A, µ) be a measure space and let f, g L 1 (X, A, µ, R). min(f, g) L 1 (X, A, µ, R). Show that max(f, g), Exercise 29. Give Borel functions f, g : R R such that f and g are Lebesgue integrable but fg is not Lebesgue integrable. Exercise 30. Let (X, A, µ) be a measure space and let f, g : X [, + ] be integrable and let h : X [, + ] be an A-measurable function that satisfies h(x) = f(x) + g(x) for µ-almost every x X. Prove that h is integrable and hdµ = fdµ + gdµ. Exercise 31. Let (X, A, µ) be a measure space and let f : X [, + ] be an A-measurable function whose integral exists and is not equal to. Show that if g : X [, + ] is an A- measurable function that satisfies f g µ-almost everywhere, then the integral of g exists and satisfies fdµ gdµ. Exercise 32. Let (X, A, µ) be a measure space, m 1 and f : X R m be measurable. If f is integrable and A A, then χ A f is measurable and integrable and define fdµ = χ A A fdµ. If f is integrable, then for every ϵ > 0 there is a δ > 0 such that A A and µ(a) < δ = fdµ < ϵ. A Exercise 33. Let X be a set, A an algebra of subsets of X, µ a countably additive measure on σ X (A) and f and g integrable vector-valued functions on the measure space (X, σ X (A), µ). Suppose that A fdµ = A gdµ for all A A. Prove that A fdµ = A gdµ for all A σ X(A) and that f = g a.e. 9

11 Exercise 34. Let f be a nonnegative Lebesgue measurable function on [a, b] [, + ]. If b < +, prove that b f(x)dx = lim b ϵ a ϵ 0+ f(x)dx. On the other hand, if b = +, prove that a b f(x)dx = lim c a c + f(x)dx. A similar statement is true if f is vector-valued and integrable, e.g., if f is C-valued and a integrable. Exercise Prove that lim a 0+ (2x sin( 1 ) 2 cos( 1 ))dx exists but the integrand is not Lebesgueintegrable over 0, a x 2 x x 2 1]. Exercise 36. Let X = R, A the Lebesgue measurable subsets of R, µ Lebesgue measure on R (so dµ = dx) and f L 1 (X, A, µ, C). Define ˆf(k) = + f(x) exp(ikx)dx for k R. Prove that ˆf : R C is well defined and continuous. Exercise 37. Let f : [0, + C be measurable and satisfy the estimate f(t) C exp(at) for some fixed C > 0 and a R and all t [0, +. Define the Laplace transform L(f)(s) = + f(t) exp( st)dt with domain Ω = {z C R(z) > a}. Show that L(f)(s) is well 0 defined on its domain and that the complex derivative L(f) (s) exists on the domain of the Laplace transform and equals + ( t)f(t) exp( st)dt. This will prove that s L(f)(s) is 0 complex analytic and therefore will vanish identically if the set of its zeroes has a limit point in its domain of definition. This fact is crucial for proving that the Laplace transform is one-to-one. Exercise 38. Let (X, A, µ) be a finite measure space and h : X R a measurable function. Prove that exp(ith(x))dµ(x) exists for t R and is continuous as a function of t. Prove that this function of t is differentiable provided h is integrable and rigorously find its derivative. Exercise 39. Suppose f : [0, 1] C is Lebesgue-integrable and that x f(y)dy = 0 for all x [0, 1]. Prove 0 that f = 0 a.e. Is the same true if it is only assumed that the integral vanishes for rational values of x? Exercise 40. Let X and Y be topological spaces with a countable basis. Then B(X Y ) σ X Y (B(X) B(Y )). In particular, f : X Y R is measurable with respect to σ X Y (B(X) B(Y )) if f is a Borel function. 10

12 Exercise 41. Let X and Y be sets, = A P(X) such that there exists {A n } n 1 A that satisfies n 1 A n = X and = B P(Y ) such that there exists {B n } n 1 B that satisfies n 1 B n = Y. Then σ X Y (σ X (A) σ Y (B)) = σ X Y ({A B A A and B B}) In particular if X and Y are topological spaces, then σ X Y (B(X) B(Y )) = σ X Y ({U V U open in X and V open in Y }) and therefore σ X Y (B(X) B(Y )) B(X Y ). In addition, if both X and Y have a countable basis, then σ X Y ({U V U open in X and V open in Y }) = σ X Y (B(X) B(Y )) = B(X Y ). Exercise 42. Let X and Y be sets, A an algebra of subsets of X, B an algebra of subsets of Y, A σ X (A), and B σ Y (B). Then A B σ X Y (A B) Furthermore, if µ a countably additive measure on A such that (X, σ X (A), µ) is σ-finite and ν a countably additive measure on B such that (Y, σ Y (B), ν) is σ-finite, then (µ ν)(a B) = µ(a)ν(b). Exercise 43. Define φ(α) = + exp( (x+iα)2 )dx for α R. Prove that φ is well defined and rigorously calculate φ (α). Exercise 44. Let (X, A, µ) be a measure space and f and f n (n 1) be C-valued measurable functions on X. If {f n } n 1 converges to f in measure and there exists an integrable function g 0 on X such that f n g a.e., then {f n } n 1 converges to f in mean and therefore f X ndµ fdµ. This exercise is not true if there is no such g even if µ is a finite measure. X Exercise 45. Let (X, ) be a semi-normed linear space and let N = {x X x = 0}. Then N is a vector subspace of X. Let X be the quotient vector space X/N and let x x be the natural linear quotient mapping. Define x = x. Show that : X [0, + is well defined and is a norm on X. Show that ( X, ) is a complete normed linear space if and only if (X, ) is a complete semi-normed linear space. If X = L 1 (X, A, µ, C), then we define L 1 (X, A, µ, C) = L1 (X, A, µ, C). 11

13 Exercise 46. If (X, A, µ) is a measure space then the collection C of locally measurable subsets of X is a σ-algebra of subsets of X that contains A. If B C and µ (B) = lub{µ(a) A A, A B and µ(a) < + }, then µ is a countably additive measure on the locally measurable subsets of X. µ (A) = µ(a) for all A A that are σ-finite under µ. If B C, then µ (B) = 0 if and only if B is locally null. If f : X C is C-measurable and {x X f(x) 0} A and is σ-finite under µ, then f is A-measurable. In particular, if f : X C is A-measurable and {x X f(x) 0} A and is σ-finite under µ and g : X C is C-measurable, then f g is A-measurable. Exercise 47. Let 0 < p < 1, f, g L p (X, A, µ, C) and define d(f, g) = f g p p = X f g p dµ. Then d(, ) is a complete translation invariant pseudo-metric on L p (X, A, µ, C). (L p (X, A, µ, C) has the peculiar property that the only continuous linear functional on L p (X, A, µ, C) is 0.) Exercise 48. If V is an inner product space, show that the mapping V V C, (x, y) x, y is continuous. Show that this mapping is uniformly continuous on bounded subsets of V V. Exercise 49. Let H be a Hilbert space and T L(H). Then T = lub{ T (x), y x, y H, x 1 and y 1}. Exercise 50. Work out Bessel s Inequality for V = C([0, 2π]), f, g = 2π f(x)g(x)dx, e 0 n (x) = exp(inx)/ 2π and f(x) = x 2. Assume the equality n 1 in the inequality of Lemma??. 1 n 2 = π2 6 Exercise 51. Use Parseval s Theorem applied to the complete orthonormal system {exp(inx)/(2π) (1/2) n Z} in L 2 ([0, 2π]) to obtain amusing identities. For example, π 2 /6 = n 1 1/n2 and π 4 /90 = n 1 1/n4. Exercise 52. If H is a Hilbert space, K a closed subspace of H, x H, and y K is the orthogonal projection of x onto K, then x y = glb z K x z. Exercise 53. Let H be a Hilbert space and = S H. The orthogonal complement of S, denoted by S, is defined to be {x H x, y = 0 for all y S}. Show that S is a closed subspace of H and that (S ) is the closed subspace of H generated by S. 12

14 Exercise 54. Let (X, A, µ) be a measure space, f, g : X [0, + ] measurable and ν(a) = fdµ. Then A ν is a countably additive measure and gdν = fgdµ. Definition 1 (Dynkin). Let X be a set and let D P(X). D is called a π-system if A, B D = A B D. D is called a λ-system (or d-system or Dynkin system) if: (i) X D; (ii) A, B D and A B = B \ A D; (iii) if {A n } n 1 D, A n A n+1 for n 1 = n 1 A n D. Exercise 55. Let X be a set and let D be a Dynkin system on X. Then: (i) D; (ii) A D = A c D; (iii) A, B D, A B = = A B D. Exercise 56 (Dynkin). Let X be a set and let D P(X). D is a Dynkin system if and only if: (i) X D; (ii) A D = A c D; (iii) if {A n } n 1 D are pairwise disjoint, then n 1 A n D. Give an example to show that a Dynkin system need not be an algebra let alone a σ-algebra. Hint: Let X have four elements. Exercise 57. Let X be a set and let D P(X) be a Dynkin system on X. Then D is a monotone family. Definition 2. If = P P(X), let D(P) be the Dynkin system generated by P, i.e., D(P) is the intersection of all Dynkin systems on X that contain P, one of which is P(X). D(P) is the smallest Dynkin system that contains P. Exercise 58 (Dynkin s π-λ Theorem). Let X be a set and let = P D P(X). If D is a Dynkin system and P is a π-system, then σ(p) D. Exercise 59. Let (X, A) be a measurable space and let µ and ν be two measures on (X, A) such that µ(x) = ν(x) < +. If P is a π-system on X such that σ(p) = A and µ(a) = ν(a) for all A P, then µ = ν. An example of such is X = R, A = B(X) and P is the set of closed bounded subintervals of X. 13

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