Introduction. Let C(S) be the ring of continuous real valued functions on a compact Hausdorff space S. Stone [5] shows that each bounded subset of C(S) has a least upper bound (in C(S)) if and only if the closure of each open subset of S is open ; in this event we call C(S) a Stone algebra. Throughout this paper C(S) is a Stone algebra. It is convenient to adjoin an object +00, not in C(S), and...

Abstract Let Ω be a compact Hausdorff space, let E be a Banach space, and let C(Ω, E) stand for the Banach space of all E-valued continuous functions on Ω under supnorm. In this paper we study when nuclear operators on C(Ω, E) spaces can be completely characterized in terms of properties of their representing vector measures. We also show that if F is a Banach space and if T : C(Ω, E) → F is a ...

We show that computability of the Radon-Nikodym derivative of a measure μ absolutely continuous w.r.t. some other measure λ can be reduced to a single application of the non-computable operator EC, which transforms enumeration of sets (in N) to their characteristic functions. We also give a condition on the two measures (in terms of the computability of the norm of a certain linear operator inv...

The representation of linear operators, on the Banach space of Bochner integrable functions, has been the object of much study for the past fifty years. Dunford and Pettis began this investigation in 1940 with the representation of weakly compact and norm compact operators on L1(R) by a Bochner integral, see [6,8]. Andrews has extended their study to the case of the space L1(E), of E-valued, Bo...

We study heat kernel measures on sub-Riemannian infinitedimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give L-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection g...

Example 1. Change of measure on Ω = [0, 1]. Example 2. Change of measure for normal random variables. • We can also perform change of measure for a whole process rather than for a single random variable. Suppose that there is a probability space (Ω,F) and a filtration {Ft, 0 ≤ t ≤ T}. Suppose further that FT -measurable Z is an almost surely positive random variable satisfying E[Z] = 1. We can ...

Proof of Theorem 1. We first establish the following Lemma. Lemma 1. Consider random variables X ∈ R and Y ∈ R. Let f Y |X be the Radon-Nikodym derivatives of probability measure P θ Y |X with respect to arbitrary measures QY provided that P θ Y |X QY . θ ∈ R is a parameter. f Y is the Radon-Nikodym derivatives of probability measure P θ Y with respect to QY provided that P θ Y QY . Note that i...

Let AI, be commutative semlslmple Banach algebras and 1 02 A2 be their projective tensor product. We prove that, if 10 2 is a group algebra (measure algebra) of a locally compact abelian group, then so are A 1 and A2. As a consequence, we prove that, if G is a locally compact abelian group and A is a comutatlve semi-simple Banach algebra, then the Banach algebra LI(G,A) of A-valued Bochner inte...