1 Questions from Silva
1.1 Section 4.5
Problem 2.
Problem 3.
1.2 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 nX−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 nX−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 nX−1 i
= lim IA∩B(T (x)) n→∞ n i=0
= Problem 3.
1.3 Section 5.2
Problem 2.
Problem 4.
Problem 5.
1
