Markov operators and invariant Baire functions
Robert E. Atalla · Rocky Mountain Journal of Mathematics · 1983
Let X be a compact r 2 -space and Ta Markov operator on C(X), i.e., T ^ 0 and T\ = 1.We deal with Baire functions which are almost everywhere T-invariant, and focus on conditions under which such Baire functions belong to the smallest system of Baire functions containing the continuous invariant functions, and closed under almost everywhere convergence.Our main result is that this occurs precisely when both T and T* are strongly ergodic.We also show that almost everywhere invariant Baire functions separate the invariant probabilities, although everywhere invariant Baire functions need not. Introduction.We define F(T) = {/in C(X): Tf = /}, F(T*) = {m in C(X)*: T*m = m}, and P(T*) = F(T*) f] probabilities.T is called strongly ergodic if for each / in C(X), T n f converges in the Banach space C(X), whereRecall that T is s.e. if and only if F{T) separates F(T*) if and only if F(T) separates P(T*) [7, Theorems 2.2 and 2.7].Let B be the set of Baire functions, so B = [}{B a : a < CÜI}, where o)\ is the first uncountable ordinal and B a the #-th Baire class.We define B(F(T)) to be the smallest set of bounded functions containing F(T), and closed under bounded pointwise sequential convergence.If / and g are in B, we say f = g i ?(T*)-ae if / = g m-ae for all m in P(T*).Let B{F{T)) a be the smallest set of bounded Baire functions containing F(T), and closed under P(r*)-ae convergence.It will be convenient to extend the operator T to an operator (again called T) on the Baire functions by letting Tf(x) = $fd(T*d x ), where d x is the Dirac measure at x.An easy transfinite induction over the Baire classes shows that if/is in B, then so is Tf.The same argument shows that if/is in B and m in F(T*), then \fdm = J Tfdm.We shall need the following known result.THEOREM.[1, Proposition 2.1].The following are equivalent.(i) Both T and T* are strongly ergodic.(ii) F(r**) =