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Fundamentals of Measure Theory, Problem Set 6 1. Let Μ, Ν Be Finite Borel Measures on R, and Let Μ × Ν Be the Product Measu

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Fundamentals of Measure Theory, Problem Set 6 1. Let Μ, Ν Be Finite Borel Measures on R, and Let Μ × Ν Be the Product Measu Fundamentals of Measure Theory, Problem Set 6 1. Let µ, ν be finite Borel measures on R, and let µ × ν be the product measure on R2. The convolution of µ and ν is the measure on R, denoted µ ∗ ν, defined by µ ∗ ν(E) = µ × ν(f(x; y) 2 R2 j x + y 2 Eg). Show that: (a) µ ∗ ν is a measure. (b) The convolution operation ∗ is commutative and associative. What measure func- tions as the neutral element for this operation? (that is, µ ∗ ν = ν for any ν?). (c) Denote by δa the Dirac measure at a (that is, the measure of a set is 1 if a is in the set, 0 otherwise). What's δa ∗ δb? Z Z (d) Show that if f is bounded and measurable then fdµ∗ν = f(x+y)dµ×ν(x; y) 2 R R (hint: first show it for f = χE). (e) Deduce from the previous item that µ[∗ ν =µ ^ · ν^ (whereµ ^ is the Fourier transform - remember that ei(x+y) = eixeiy). Remarks: • The concept of convolution of measures is also connected to probability, as follows. Suppose (Ω; B; µ) is a probability space, and suppose f is a random variable. The probability distribution of f is defined to be the push-forward measure f∗µ (which is a Borel measure on R). If f and g are independent random variables, then one can show that (f + g)∗µ = f∗µ ∗ g∗µ (the probability distribution of the sum is the convolution of the probability distributions). In turn, this means that the Fourier transform of the probability distribution of the sum is the product of the Fourier transforms of the two probability distributions. This observation makes Fourier techniques useful in probability. • We defined the convolution of measures on R to be concrete, however this could be defined in greater generality for other groups (we only used the + operation for R here). Suppose (Ω; B) is a measurable space, such that Ω is also a group, and the group operations (g; h) 7! gh from Ω × Ω ! Ω and g 7! g−1 from Ω ! Ω are measurable. Then one could define convolution of measures in a similar way. This is typically of interest when we are dealing with topological groups and their representations. 2. Let (Ωi; Bi; µi) be a sequence of probability spaces, and let (Ω; B; µ) be the infinite product probability space. Show that for any E 2 B and any " there exists a set F 2 B which depends on finitely many coordinates such that µ(E4F ) < " (4 denotes the symmetric difference). 3. Consider f0; 1g with the probability measure which assigns to each point the value 1=2. Consider the space Ω = f0; 1gZ (the infinite product indexed by Z), with the product measure µ. Let S :Ω ! Ω denote the map which shifts all the indices of a point one spot to the right (or the left, if you prefer). It is easy to check that S preserves the measure, that is, µ(S(E)) = µ(E) (check it if it is not obvious). Sn will denote S applied n times (i.e. shifting the coordinates n spots). (a) Show that if E; F 2 B depend on finitely many coordinates then there exists some N such that for any n ≥ N we have that µ(E \ Sn(F )) = µ(E)µ(F ). 1 (b) Deduce that if E; F 2 B are any measurable sets then for any " there exists N such that for any n ≥ N we have that jµ(E \ Sn(F )) − µ(E)µ(F )j < " (hint: approximate the sets E; F by sets which depend on finitely many coordinates, as in the previous problem set). (c) Deduce that if E is S-invariant, that is S(E) = E, then either µ(E) = 0 or µ(E) = 1. A measure preserving transformation for which every invariant set has either measure 0 or measure 1 is called an ergodic transformation. (d) You can identify f0; 1gZ with f0; 1gN × f0; 1gN (the first copy will be the negative coordinates, and the other the non-negative ones), and up to measure zero you can identify f0; 1gN ×f0; 1gN with [0; 1]2 (by sending each infinite sequence of zeros and ones the number which has it as its binary expansion). Under this identification, consider at the set [0; 1=2]2, and sketch its images under S; S2;S3;S4. 4. Suppose µ, ν are finite measures, and let ρ = µ − ν. Show that there is a finite measure λ such that µ = ρ+ + λ, ν = ρ− + λ. 5. Let µ be Lebesgue measure on R, and let ν be counting measure on R (with the Borel σ-algebra). Show that ν does not admit a Lebesgue decomposition with respect to µ. 6. An atom of a measure ν is a measurable set A such that ν(A) > 0 and for any measurable subset B ⊆ A, either ν(A) = ν(B) or ν(B) = 0. (a) Suppose ν is a Borel measure on R which assigns to finite intervals finite measure. Show that if A is an atom of ν, then there is a point x 2 A such that fxg is an atom for ν. (b) We say that a measure ν atomic if each measurable set can be written as a disjoint union of atoms (along with a set of measure zero). Suppose µ is a finite measure on [0; 1]. Show that we can decompose µ = µ1+µ2+µ3 such that each is singular with respect to the other, µ1 is absolutely continuous with respect to Lebesuge measure, µ2 is singular with respect to Lebesgue measure and has no atoms, and µ3 is an atomic measure. dν 7. Suppose µ, ν are σ-finite measrures on (Ω; B), and ν µ. Let ρ = µ + ν, and f = dρ , dν f then µ(f! j f(!) 62 [0; 1)g) = 0, and dµ = 1−f (which is well defined almost everywhere with respect to µ, because f is less than 1 almost everywhere). 8. Let (Ω; B; µ) be a probability space, and let B0 be a sub-σ-algebra of B. Let EB0 : 1 1 L (Ω; B; µ) ! L (Ω; B0; µ) be the conditional expectation. Suppose f ≥ 0 is a bounded 2 2 function which is measurable with respect to B. Show that if EB0 (f ) = EB0 (f) then f = EB0 (f) almost everywhere (with respect to B). (In particular, in that case, f coincides up to a set of measure zero with a function which is measurable with respect to B0). 2.
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