POSITIVE OPERATORS ON C(X)

Conditions for the existence of finite and a finite invariant measures for a Markov operator on C(X) are studied. Let X be a locally compact Hausdorff space and 2 be the collection of Baire sets. Let P be an operator on C(X) which satisfies (1) ||-P||=1. (2) Pl = l. Now if/ = 0 then PfS^O, because we may assume that 0^/gl and thenP(l-/) = l-P/gl. We shall also assume: (3) If p is a countably additive measure then so is P*p. By a measure we mean a positive finite measure, unless otherwise stated. Every countably additive measure is regular since it is defined on Baire sets. It is enough to assume (3) only for ju = 8X, x£X, since then P(x, A) = P*èx(A) is a probability measure for a fixed x and if .4 £2, P(-, A) is 2 measurable: If 4 £2 is compact then there exists a sequence of continuous functions 0^/^ = 1,/« J, 1a (where l¿(x) =0 if x(£A and 1 if xEA). Thus P(x, ^4)=lim„P*5I(fn)=limnP/„0) and is measurable. Now the collection of AE^ such that P(-, A) is measurable is a a field and thus all of 2. Clearly P*n(A) =fP(x, A)n(dx) is a countably additive measure whenever ¡j. is. Also P/O) =fP(x, dy)f(y) and if 0 5¡/„s;i and/n J,/ then Pfn(x)—*Pf(x). This last property almost implies (3). Theorem 1. Let Xbea locally compact Hausdorff space and X = U Cn where C„ are compact Baire sets. If the operator P satisfies (1), (2) and (3'): (3') If 0 £/» Ú1, /« If and /., fE C(X) then Pfn(x)^P/(x). Then condition (3) holds. Proof. The operator P* is defined on finitely additive regular measures, by [l, Theorem IV.6.2]. Let 0^/u be countably additive and put P*p = v0-\-vi as in [4, p. 52]; namely, vi is countably additive and vo is purely additive, i.e. if 0=Xi£i'o and X is countably additive then X = 0. Now if A £2 is compact the restriction of v0 to A is countably additive by [l, Theorem III. 5.13] and thus v0(A)=0. Hence Received by the editors August 30, 1968. 1 Supported by the National Science Foundation Grant GP-7475.