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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