[SOLVED] Dynamical Homework 11

1.1            Section 5.1

Problem 2.

Let T : X → X be a measure-preserving transformation with µ(X) = 1 and suppose that for every measurable set A the limit

1 n_X−1            _i

lim           IA(T (x))

n→∞ n i=0

exists and equals µ(A) a.e. We want to show that T is ergodic. Let us fix two measurable sets A,B. Other than on two sets of measure 0 (whose union is measure 0), we have,

1 n_X−1            _i

µ(A)µ(B) = lim  _A  IB(T (x)) n→∞ n i=0 n→∞ n i=0

1 nX−1nX−1            i                     j

= lim                    IA(T (x))IB(T (x))

ⁿ→∞ n _i=0 j=0

1 n_X−1                 _i

= lim  IA_∩B(T (x)) n→∞ n i=0

= Problem 3.

1.2            Section 5.2

Problem 2.

Problem 4.

Problem 5.

1