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The concepts of accretive and differentiable operator in a Banach space B are used to show that certain approximations to a solution of a nonlinear evolution equation converge. When B is a space of continuous functions it is shown that the approximations and the solution be represented as integrals with respect to a signed measure on a function space. As an example, a new proof is given for the existence and uniqueness of solutions to a nonlinear parabolic differential equations with coefficients dependent upon solutions. Integral representations of these solutions follow.

In this paper, we shall show that any Gaussian measure on a separable or reflexive Banach space is an abstract Wiener measure in the sense of L. Gross [1] and, for the proof of that, establish the Radon extensibility of a Gaussian measure on such a Banach space. In addition, we shall give some remarks on the support of an abstract Wiener measure.

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .

Let $\mathscr{P} = \{F_0,\cdots, F_m\}$ be a class of probability measures on $(\mathscr{X}, \mathscr{B})$. For any signed measure $\tau$ on $\mathscr{B}^N$, let $\tau^\ast$ be the average of $\tau g$ over all $N$! permutations $g$ and let $\|\tau\| = \mathbf{V}\{|\tau(C)|: C \in \mathscr{B}^N\}$. Let $d_{ij} = \|F_i - F_j\|$ and $K(x) = .5012\cdots x(1 - x)^{-\frac{3}{2}}$. For any nonnegative integral partitions $\mathbf{N} = (N_0,\cdots, N_m)$ and $\mathbf{N}' = (N_0',\cdots, N_m')$ of $N$, let $\delta_i = N'_i - N_i$ and $\wedge_i = (N'_i \wedge N_i) + 1$. With $\tau = \times F_i^{N_i} - \times F_i^{N_i'}$ and $n = {\tt\#}\{i\mid\delta_i \neq 0\} - 1$, we bound $\|\tau^\ast\|^2$ by \begin{equation*}\tag{T3} nK(d)\sum \delta_i^2\wedge_i^{-1}\quad \text{with} d = \vee\{d_{ij}\mid\delta_i \neq 0, \delta_j \neq 0\}\end{equation*} and, if $\mathscr{P}$ is internally connected by chains with non-orthogonal successive elements, by \begin{equation*}\tag{T4} \frac{1}{2}mK(\check{d})(\sum |\delta_i|)^2(\sum \wedge_i^{-1}) \quad\text{with} \check{d} = \vee\{d_{ij} \mid F_i ?? F_j\}.\end{equation*} The bound (T3) is finite iff the $F_i$ with $\delta_i \neq 0$ are pairwise non-orthogonal and (T4) is designed to replace it otherwise.

Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an "anti-lattice"1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP\, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2

It is well known that certain results such as the Radon-Nikodym Theorem, which are valid in totally σ -finite measure spaces, do not extend to measure spaces in which μ is not totally σ -finite. (See §2 for notation.) Given an arbitrary measure space (X, S, μ) and a signed measure ν on (X, S), then if ν ≪ μ for X, ν ≪ μ when restricted to any e ∊ Sf and the classical finite Radon-Nikodym theorem produces a measurable function ge(x), vanishing outside e, with

We shall consider plane Borel sets of points, though the result and the proof hold in space of any number of dimensions and for more general sets.

in a substrip R, x (to , h), where 01 is a signed measure and where the integral converges absolutely. We further show that if a function U(X, t) is representable in the form (2) in a strip R, x (t, , T) and if the integral converges absolutely, then U(X, t) satisfies (1) in a substrip Ii, x (2, , h). These results will extend the earlier work of Tychonoff [l], Krzytariski [2], [3], Widder [4], Rosenbloom [5], and others, and the more recent work of Friedman [6], Krzyianski [7], and Aronson [8]. We then use these results to extend a uniqueness theorem of Krzyiariski [7], which in turn extended a uniqueness theorem of Widder [4], and also to extend a recent uniqueness theorem of Aronson [9]. This uniqueness theorem, Theorem 9, is a direct extension of a result of Rosenbloom [5].

FN'(t) = inf {x:FN(x) > t}; and we write g o FN-l for the composed function g (FN-1). Note that TN = aO g o FN-1 dVN when the signed measure VN puts mass CNi/N at i/N for i = 1, * * , N and puts 0 mass elsewhere. Let v denote a signed measure on (0, 1). (The signed measures VN will not be used, but v is in some sense their limit.) For technical reasons to be displayed below, we bound ourselves away from 0 and 1 by an amount #N where NiN -3 0 as N - oo at a rate to be specified later. Let (1.2) 1N - fANd (1.2) Y~~~~~~.N = fAN g dv

1. Notation. We shall use the notation of [1] for topological and measure theoretic concepts. Let X be a normal topological space. Let P(x, A) be the transition probabilities of a Markov Process: 1.1. For a fixed xGX the set function P(x,) is a measure, on the Borel sets, of total measure one. 1.2. For a fixed Borel set A, the function P(., A) is Borel measurable. By a measure we shall mean a countably additive positive measure, unless otherwise stated. Let us denote by r b a the set of regular bounded finitely additive signed measures on X and by r c a those elements of r b a which are countably additive. The transition probabilities induce an operator on the bounded measurable functions by 1.3. (Pf) (x) =ff(y)P(x, dy). Also if j. is a bounded finitely additive signed measure one defines 1.4. (juP) (A) =fP(x, A),ju(dx). It is well known that 1.5. f(Pf) (x) A (dx) =ff(x) (juP) (dx) and that ,uP is countably additive if A is. Throughout the paper we assume: 1.6. If fE C(X) then PfE C(X), where C(X) denotes the continuous functions. Also: 1.7. If .lEr ca then APCrc a. These two conditions are always satisfied under the assumptions of [2 ]: under the assumptions of [2 ] every countably additive measure is regular. Another example is given by (PD (x) =f(?>(x)) where 4, is a homeomorphism of X onto X.

Publisher Summary This chapter discusses the extremal structure of the range of a controlled vector measure v with values in a Hausdorff locally convex space X over the field of reals. The chapter also presents the determination of extreme points of a closed convex null of the range of v in terms of levels of the weak integral map. These points are strongly extreme (denting) when Hausdorff locally convex space is quasi-complete. The chapter presents a theorem that relates the existence of an exposed point of the range with the existence of a continuous linear functional on Hausdorff locally convex space for which the signed measure is equivalent to controlled vector measure. The range of v, its weak closure, and the closed convex hull have the same exposed points, which in turn are strongly exposed.

Let {X j ; j = 1, 2,...} be a finite sequence of independent random variables. Let S = Σ X j be their sum, and let P j be the distribution of X j . Let M be the measure defined on the line deprived of its origin by M (A) = Σ j P j {A ∩ {0}c}. The purpose of the present paper is to develop certain results on the approximation of the distribution L (S)of S by the accompanying infinitely divisible distribution which has for Paul Levy measure the measure M itself. If λ= || M || is the total mass of M then V = M/λ is a probability measure. Let {Z k ; k = 1, 2,...} be an independent sequence of random variables having common distribution V. Let N be a Poisson variable independent of the Z k and such that EN = λ. A “natural” infinitely divisible approximation to the distribution of S is the distribution of T = Zk with Z o = 0. If μ is a signed measure, let || μ || be its norm, equal to the total mass || μ || = ||μ + || + || u − ||. It can be shown that in some cases the approximation of L (S)by L (T)is good in the sense that || L (S) — L (T) || is small. More generally it will be shown that the Kolmogorov-Smirnov distance ϱ [L (S), L (T)] is small. This distance is defined by $$ \left( \mu ,\upsilon \right)=\sup \left| \mu \left\{ \left( -\infty ,x \right] \right\}-\upsilon \left\{ \left( -\infty ,x \right] \right\} \right| $$ for any two signed measures,u and v. One could also use Paul Levy’s diagonal distance Λ [μ, v] defined as the infimum of numbers α such that $$ v\left\{ \left( -\infty ,x-\alpha \right] \right\}-\alpha \le \mu \left\{ \left( -\infty ,x \right] \right\}\le v\left\{ \left[ -\infty ,x+\alpha \right] \right\}+\alpha $$ for every value of x. However, since Λ is not invariant under scale changes, approximations in this sense are not always entirely satisfactory.

Let $X_1, X_2, \cdots$ be independent identically distributed random variables having a common distribution function $F$ and let $f_n(x) = (na_n)^{-1} \sum^n_{i = 1} k((x - X_i)/a_n)$ where $\{a_n\}$ is a sequence of positive numbers converging to zero and $k$ is a probability density function. If $\sum^\infty_{n = 1} \exp(- cna_n^2)$ is finite for all positive $c$ and if $k$ satisfies: (i) $k$ is continuous and of bounded variation on $(-\infty, \infty)$. (ii) $uk(u) \rightarrow 0$ as $u \rightarrow + \infty$ or $-\infty$. (iii) There exists a $\delta$ in (0, 1) such that $u(V^{-u^\delta}_{-\infty} (k) + V^\infty_{u^\delta} (k)) \rightarrow 0$ as $u \rightarrow \infty$. (iv) $\int|u| dk(u)$, the integral of $|u|$ with respect to the signed measure determined by $k$, is finite. Then the author [2] has established the following: THEOREM. A necessary and sufficient condition for $\lim_{n\rightarrow\infty} \sup_x|f_n(x) - g(x)| = 0$ with probability one for a function $g$ is that $g$ be the uniformly continuous derivative of $F$. The purpose of this note is to show that this theorem remains true if conditions (i)-(iv) on $k$ are replaced by the condition that $k$ is of bounded variation on $(-\infty, \infty)$.

The purpose of this paper is to give a necessary and also a sufficient condition for a Dirichlet finite harmonic function on a Riemann surface to be represented as a difference of a Dirichlet finite solution of Au=Pu (P_O) and a Dirichlet finite potential of signed measure. 1. Let P=P(z) dx dy (z=x+iy) be a nonnegative not identically zero o-Holder continuous (0<1) second order differential on a Riemann surface R and PD(R) be the Hilbert space of all Dirichlet finite solutions of (1) AU(Z) = P(Z)U(Z), A = 4a2./I aZa, on R with the scalar product given by mixed Dirichlet integral, i.e. (u, v)=D11(u, v)=fS11 duA*dv, not the energy integral. The study of PD(R) was begun by Royden [6]. We will use the fact shown by Nakai [2] that PD(R) forms a vector lattice under the natural order in PD(R). We also use the Glasner-Katz maximum principle [1] that the modulus of every function in PD(R) takes its maximum on the Royden harmonic boundary. The recent result of Nakai [3] that PBD(R) is dense in PD(R) will not be made use of. Let A(R) be the Royden harmonic boundary and HD(R) be the class of Dirichlet finite harmonic functions on R. (For the basic materials from the Royden compactification and the class HD(R) we refer to the monograph of Sario and Nakai [7].) One of the important problems in the theory of PD(R) which is not fully developed yet is to describe the distribution of PD(R)|A(R) in HD(R)|A(R). We will prove a theorem which contributes to this question. 2. If R is parabolic, then PD(R)= {0} (cf. Royden [6]), which case offers no interest. Therefore we assume throughout the paper that R is hyperbolic. Let M(R) be the class of all Dirichlet finite Tonelli functions on R and MA(R) the subclass of M(R) consisting of functions f with f JA(R)=O (cf. [7]). We then have the orthogonal decomposition M(R) = HD(R) + MA(R), Received by the editors May 26, 1971. AMS 1970 subject classifications. Primary 31A05, 58G99. 1 The work was sponsored by the U.S. Army Research Office-Durham, Grant DA-ARO-D-31-124-71-G20, UCLA. (e, American Mathematical Society 1972