Homework Assignment 5 - BYU Mathtfisher/documents/classes/2013/winter/643/... · Homework...
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Homework Assignment 5 February 21, 2013 1. Let T : X → Y be continuous. Show T induces a natural map M (X ) → M (Y ) and that this map is continuous in the weak * topology. 2. Let α ∈ (0, 1) be irrational and T : T d → T d be defined by T (x 1 , ..., x d )=(x 1 + α, x 2 + a 21 x 1 , ..., x k + a k1 x 1 + ··· a kk-1 x k-1 ) where the coefficients a ij are integers and a ii-1 6= 0. Use a Fourier transformation to show that T is ergodic with respect to Lebesgue. (Start with low dimension if this is easiest). 3. Let (X, A,μ) and (Y, B,ν ) be measure spaces and T : X → Y be measure preserving. Prove that U T is surjective if and only if ˜ T : ˜ B→ ˜ A is an isomorphism. 4. For a probability space (X, A,μ) define a metric on ˜ A by d(A, B)= μ(A 4 B). (a) Prove that d(·, ·) is indeed a metric and that with this metric the set ˜ A is complete. (b) Prove that (X, A,μ) is separable if and only if ˜ A is separable. 1
Transcript of Homework Assignment 5 - BYU Mathtfisher/documents/classes/2013/winter/643/... · Homework...
Homework Assignment 5
February 21, 2013
1. Let T : X → Y be continuous. Show T induces a natural map M(X)→M(Y ) and that this map is continuous in the weak∗ topology.
2. Let α ∈ (0, 1) be irrational and T : Td → Td be defined by
T (x1, ..., xd) = (x1 + α, x2 + a21x1, ..., xk + ak1x1 + · · · akk−1xk−1)
where the coefficients aij are integers and aii−1 6= 0. Use a Fouriertransformation to show that T is ergodic with respect to Lebesgue.(Start with low dimension if this is easiest).
3. Let (X,A, µ) and (Y,B, ν) be measure spaces and T : X → Y bemeasure preserving. Prove that UT is surjective if and only if T : B → Ais an isomorphism.
4. For a probability space (X,A, µ) define a metric on A by d(A,B) =µ(A4B).
(a) Prove that d(·, ·) is indeed a metric and that with this metric theset A is complete.
(b) Prove that (X,A, µ) is separable if and only if A is separable.
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