Mean Ces\`aro summability of Laguerre and Hermite series
Eileen L.
Poiani
1-31
Abstract: The primary purpose of this paper is to prove inequalities of the type $\vert\vert{\sigma _n}(f,x)W(x)\vert{\vert _p} \leqslant C\vert\vert f(x)W(x)\vert{\vert _p}$ where ${\sigma _n}(f,x)$ is the $ n$th $(C,1)$ mean of the Laguerre or Hermite series of $f, W(x)$ is a suitable weight function of particular form, $C$ is a constant independent of $ f(x)$ and $n$, and the norm is taken over $(0,\infty )$ in the Laguerre case and $( - \infty ,\infty )$ in the Hermite case for $1 \leqslant p \leqslant \infty$. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For $p < \infty$ and $f(x)W(x) \in {L^p}$, mean summability results showing that $ \mathop {\lim }\nolimits_{n \to \infty } \vert\vert[{\sigma _n}(f,x) - f(x)]W(x)\vert{\vert _p} = 0$ are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.
$\Pi \sp{0}\sb{1}$ classes and degrees of theories
Carl G.
Jockusch;
Robert I.
Soare
33-56
Abstract: Using the methods of recursive function theory we derive several results about the degrees of unsolvability of members of certain $\Pi _1^0$ classes of functions (i.e. degrees of branches of certain recursive trees). As a special case we obtain information on the degrees of consistent extensions of axiomatizable theories, in particular effectively inseparable theories such as Peano arithmetic, ${\mathbf{P}}$. For example: THEOREM 1. If a degree $ {\mathbf{a}}$ contains a complete extension of $ {\mathbf{P}}$, then every countable partially ordered set can be embedded in the ordering of degrees $\leqslant {\mathbf{a}}$. (This strengthens a result of Scott and Tennenbaum that no such degree ${\mathbf{a}}$ is a minimal degree.) THEOREM 2. If $ {\mathbf{T}}$ is an axiomatizable, essentially undecidable theory, and if $ \{ {{\mathbf{a}}_n}\}$ is a countable sequence of nonzero degrees, then ${\mathbf{T}}$ has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each ${{\mathbf{a}}_n}$. THEOREM 3. There is a complete extension $ {\mathbf{T}}$ of ${\mathbf{P}}$ such that no nonrecursive arithmetical set is definable in $ {\mathbf{T}}$. THEOREM 4. There is an axiomatizable, essentially undecidable theory $ {\mathbf{T}}$ such that any two distinct complete extensions of ${\mathbf{T}}$ are Turing incomparable. THEOREM 5. The set of degrees of consistent extensions of ${\mathbf{P}}$ is meager and has measure zero.
Automorphisms of $\omega \sb{1}$-trees
Thomas J.
Jech
57-70
Abstract: The number of automorphisms of a normal $ {\omega _1}$-tree $ T$, denoted by $\sigma (T)$, is either finite or ${2^{{\aleph _0}}} \leqslant \sigma (T) \leqslant {2^{{\aleph _1}}}$. Moreover, if $ \sigma (T)$ is infinite then $\sigma {(T)^{{\aleph _0}}} = \sigma (T)$. Moreover, if $T$ has no Suslin subtree then $\sigma (T)$ is finite or $\sigma (T) = {2^{{\aleph _0}}}$ or $\sigma (T) = {2^{{\aleph _1}}}$. It is consistent that there is a Suslin tree with arbitrary precribed $\sigma (T)$ between ${2^{{\aleph _0}}}$ and ${2^{{\aleph _1}}}$, subject to the restriction above; e.g. $ {2^{{\aleph _0}}} = {\aleph _1},{2^{{\aleph _1}}} = {\aleph _{324}}$ and $\sigma (T) = {\aleph _{17}}$. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen's method of forcing and Jensen's techniques in $L$.
Iterated fine limits and iterated nontangential limits
Kohur
Gowrisankaran
71-92
Abstract: Let ${\Omega _k},k = 1{\text{ to }}n$, be harmonic spaces of Brelot and ${u_k} > 0$ harmonic functions on ${\Omega _k}$. For each $w$ in a class of multiply superharmonic functions it is shown that the iterated fine limits of $[w/{u_1} \cdots {u_n}]$ exist up to a set of measure zero for the product of the canonical measures corresponding to ${u_k}$ and are independent of the order of iteration. This class contains all positive multiply harmonic functions on the product of ${\Omega _k}$'s. For a holomorphic function $ f$ in the Nevanlinna class of the polydisc ${U^n}$, it is shown that the $n$th iterated fine limits exist and equal almost everywhere on ${T^n}$ the $n$th iterated nontangential limits of $ f$, for any fixed order of iteration. It is then deduced that, with the exception of a set of measure zero on ${T^n}$, the absolute values of the different iterated limits of $f$ are equal. It is also shown that the $ n$th iterated nontangential limits are equal almost everywhere on ${T^n}$ for any $f$ in $ {N_1}({U^n})$.
The exceptional subset of a $C\sb{0}$-contraction
Domingo A.
Herrero
93-115
Abstract: Let $T$ be a ${C_0}$-operator acting on a (complex separable) Hilbert space $ \mathcal{K}$; i.e., $ T$ is a contraction on $\mathcal{K}$ and it satisfies the equation $ q(T) = 0$ for some inner function $q$, where $q(T)$ is defined in the sense of the functional calculus of B.Sz.-Nagy and C. Foiaş. Among all those inner functions $q$ there exists a unique minimal function $ p$ defined by the conditions: (1) $p(T) = 0$; (2) if $q(T) = 0$, then $p$ divides $q$. A vector $ F \in \mathcal{K}$ is called exceptional if there exists an inner function $r$ such that $r(T)F = 0$, but $p$ does not divide $r$. The existence of nonexceptional vectors plays a very important role in the theory of $ {C_0}$-operators. The main result of this paper says that nonexceptional vectors actually exist; moreover, the exceptional subset of a $ {C_0}$-operator is a topologically small subset of $ \mathcal{K}$.
Analytically invariant and bi-invariant subspaces
Domingo Antonio
Herrero;
Norberto
Salinas
117-136
Abstract: The purpose of this paper is to call attention to some interesting weakly closed algebras related to a bounded linear operator $ T$ acting on a Banach space $\mathfrak{X}$ and their associated lattices of invariant subspaces, namely, the algebras generated by the polynomials and by the rational functions in $ T$, and the commutant and the double-commutant of $T$. The relationship between those algebras and their lattices, as well as the ones corresponding to the operators induced by $T$ on an invariant subspace (restriction), or on the quotient space $ \mathfrak{X}/\mathfrak{M}$ (where $ \mathfrak{M}$ is an invariant subspace of a given type) are analyzed. Several results relative to the decomposition of invariant subspaces and the topological structure of the lattices (under the ``gap-between-subspaces'' metric topology) are also considered.
Localizations of HNP rings
James
Kuzmanovich
137-157
Abstract: In this paper it is shown that every hereditary Noetherian prime ring is the intersection of a hereditary Noetherian prime ring having no invertible ideals with a bounded hereditary Noetherian prime ring in which every nonzero two-sided ideal contains an invertible two-sided ideal. Further, it is shown that this intersection corresponds to a decomposition of torsion modules over such a ring; if $R$ is an HNP ring with enough invertible ideals, then this decomposition coincides with that of Eisenbud and Robson. If $M$ is a maximal invertible ideal of $ R$ where $R$ is as above, then an overring of $ R$ is constructed which is a localization of $R$ at $M$ in a ``classical sense"; that is, it is a ring of quotients with respect to a multiplicatively closed set of regular elements satisfying the Ore conditions. The localizations are shown to have nonzero radical and are also shown to satisfy a globalization theorem. These localizations are generalizations of ones constructed by A. V. Jategaonkar for HNP rings with enough invertible ideals.
On the classification of simple antiflexible algebras
Mahesh Chandra
Bhandari
159-181
Abstract: In this paper, we begin a classification of simple totally antiflexible algebras (finite dimensional) over splitting fields of characteristic $\ne 2,3$. For such an algebra $A$ let $P$ be the largest associative ideal in $ {A^ + }$ and let $ N$ be the radical of $ P$. We say that $ A$ is of type $ (m,n)$ if $N$ is nilpotent of class $ m$ with $\dim A = n$. Define ${N_i} = {N_{i - 1}} \cdot N,{N_1} = N$, then $ A$ is said to be of type $ (m,n,{d_1},{d_2}, \cdots ,{d_q})$ if $A$ is of type $(m,n),\dim ({N_i} - {N_{i - 1}}) = {d_i}$ for $1 \leqslant i \leqslant q$ and $\dim ({N_i} - {N_{i + 1}}) = 1$ for $ q < i < m$. We then determine all nodal simple totally antiflexible algebras of types $(n,n),(n - k,n,k + 1),(n - 2,n)$ (over fields of characteristic $\ne 2,3$) and of type (3, 6) (over the field of complex numbers). We also give preliminary results for nodal simple totally antiflexible algebras of type $(n - k,n,k,2)$ and of type $(m,n,{d_1}, \cdots ,{d_q})$ in general with $m > 2$ (the case $m = 2$ has been classified by D. J. Rodabaugh).
On bounded oscillation and asymptotic expansion of conformal strip mappings
Arthur E.
Obrock
183-201
Abstract: Relations between the boundary parameters ${\phi _ - },{\phi _ + }$ of a strip $S = \{ {\phi _ - }(x) < y < {\phi _ + }(x)\}$ and the values $f(x)$ of its canonical conformal mapping onto a horizontal strip $H = \{ \vert\upsilon \vert < 1\}$ are studied. Bounded oscillation $({\max _y}\operatorname{Re} f(x + iy) - {\min _y}\operatorname{Re} f(x + iy) = \omega (x) = O(1))$ is characterized in terms of ${\phi _ - },{\phi _ + }$. A formal series expansion $ \upsilon = \sum {y^m}{a_{m,n}}(x)$ is derived for the solution to the Dirichlet problem on $S$ and its partial sums are used to obtain formulas for the asymptotic expansion of $f$ in terms of ${\phi _ + },{\phi _ - }$.
On the genus of a group
Arthur T.
White
203-214
Abstract: The genus of a group is defined to be the minimum genus for any Cayley color graph of the group. All finite planar groups have been determined, but little is known about the genus of finite nonplanar groups. In this paper two families of toroidal groups are presented; the genus is calculated for certain abelian groups; and upper bounds are given for the genera of the symmetric and alternating groups and for some hamiltonian groups.
Actions of groups of order $pq$
Connor
Lazarov
215-230
Abstract: We study the bordism group of stably complex $G$-manifolds in the case where $G$ is a metacyclic group of order $ pq$ and $p$ and $q$ are distinct primes. This bordism group is a module over the complex bordism ring and we compute the projective dimension of this module. We develop some techniques necessary for the study of this module in case $ G$ is a more general metacyclic group.
The equivariant Plateau problem and interior regularity
H. Blaine
Lawson
231-249
Abstract: Let $M \subset {{\text{R}}^n}$ be a compact submanifold of Euclidean space which is invariant by a compact group $G \subset SO(n)$. When $\dim (M) = n - 2$, it is shown that there always exists a solution to the Plateau problem for $M$ which is invariant by $G$ and, furthermore, that uniqueness of this solution among $G$-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for $M$ within the class of $G$-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space ${\text{R}}/G$ where, for ``big'' groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds $M$, the cone $ C(M) = \{ tx;x \in M$ and $ 0 \leqslant t \leqslant 1\}$ is the unique solution to the Plateau problem for $ M$, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type $ {S^n} \times {S^n} \subset {{\text{R}}^{2n + 2}},n \geqslant 3$, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in $ {{\text{R}}^8}$ and examples in $ {{\text{R}}^n}$ for $n \geqslant 10$ with any prescribed Betti number nonzero.
A class of representations of the full linear group. II
Stephen
Pierce
251-262
Abstract: Let $V$ be an $n$-dimensional vector space over complex numbers $ C$. Let $W$ be the $m$th tensor product of $V$. If $T \in {\operatorname{Hom} _C}(V,V)$, let ${ \otimes ^m}T \in {\operatorname{Hom} _C}(W,W)$ be the $m$th tensor product of $T$. The homomorphism $T \to { \otimes ^m}T$ is a representation of the full linear group $ {\text{G}}{{\text{L}}_n}(C)$. If $H$ is a subgroup of the symmetric group $ {S_m}$, and $\chi$ a linear character on $ H$, let $V_\chi ^m(G)$ be the subspace of $W$ consisting of all tensors symmetric with respect to $H$ and $\chi$. Then $ V_\chi ^m(H)$ is invariant under ${ \otimes ^m}T$. Let $K(T)$ be the restriction of ${ \otimes ^m}T$ to $ V_\chi ^m(H)$. For $ n$ large compared with $ m$ and for $H$ transitive, we determine all cases when the representation $T \to K(T)$ is irreducible.
On the null-spaces of elliptic partial differential operators in $R\sp{n}$
Homer F.
Walker
263-275
Abstract: The objective of this paper is to generalize the results of Lax and Phillips [4] and Walker [6] to include elliptic partial differential operators of all orders whose coefficients approach constant values at infinity with a certain swiftness. An example is given of an elliptic operator having an infinite-dimensional null-space whose coefficients slowly approach constant limiting values.
Asymptotic behavior of linear integrodifferential systems
Viorel
Barbu;
Stanley I.
Grossman
277-288
Abstract: We consider the system $n \times n$ matrices. System $({\text{L)}}$ generates a semigroup given by ${T_t}f(s) = y(t + s;f)$ for $f$ bounded, continuous and having a finite limit at $- \infty$. Under hypotheses concerning the roots of $\det (\lambda I - A - \hat B(\lambda ))$, where $\hat B(\lambda )$ is the Laplace transform, various results about the asymptotic behavior of $y(t)$ are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If $B(t) \in {L^1}[0,\infty )$ and $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $\operatorname{Re} \lambda > 0$, then for every $\epsilon > 0$, there is an ${M_{\epsilon}}$ such that $\vert\vert{T_t}f\vert\vert \leqslant {M_{\epsilon}}{e^{\epsilon t}}\vert\vert f\vert\vert$. Theorem 2. If ${(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $\operatorname{Re} \lambda > - \alpha (\alpha > 0)$ and if $B(t){e^{\alpha t}} \in {L^1}[0,\infty )$, then the solution to $ ({\text{L)}}$ is exponentially asymptotically stable.
On the semisimplicity of group rings of solvable groups
C. R.
Hampton;
D. S.
Passman
289-301
Abstract: Let $K[G]$ denote the group ring of $ G$ over the field $ K$ of characteristic $p > 0$. An interesting unsolved problem is to find necessary and sufficient conditions on $G$ for $K[G]$ to be semisimple. Even the special case in which $G$ is assumed to be a solvable group is still open. In this paper we prove a number of theorems which may be of use in this special case.
Closed subgroups of lattice-ordered permutation groups
Stephen H.
McCleary
303-314
Abstract: Let $G$ be an $l$-subgroup of the lattice-ordered group $A(\Omega )$ of order-preserving permutations of a chain $\Omega$; and in this abstract, assume for convenience that $G$ is transitive. Let $ \bar \Omega$ denote the completion by Dedekind cuts of $\Omega$. The stabilizer subgroups $ {G_{\bar \omega }} = \{ g \epsilon G\vert\bar \omega g = \bar \omega \} ,\bar \omega \epsilon \bar \Omega$, will be used to characterize certain subgroups of $G$ which are closed (under arbitrary suprema which exist in $G$). If $\Delta$ is an $o$-block of $G$ (a nonempty convex subset such that for any $g \epsilon G$, either $\Delta g = \Delta$ or $\Delta g \cap \Delta$ is empty), and if $ \bar \omega = \sup \Delta ,{G_\Delta }$ will denote $ \{ g \epsilon G\vert\Delta g = \Delta \} = {G_{\bar \omega }}$; and the $ o$-block system $\tilde \Delta$ consisting of the translates $ \Delta g$ of $ \Delta$ will be called closed if $ {G_\Delta }$ is closed. When the collection of $o$-block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system $ \mathcal{C}$, and all systems above $ \mathcal{C}$ are closed. $\mathcal{C}$ is the trivial system (of singletons) iff $G$ is complete (in $A(\Omega )$). $ {G_{\bar \omega }}$ is closed iff $\bar \omega$ is a cut in $\mathcal{C}$ i.e., $ \bar \omega$ is not in the interior of any $\Delta \epsilon \mathcal{C}$. Every closed convex $l$-subgroup of $G$ is an inter-section of stabilizers of cuts in $\mathcal{C}$. Every closed prime subgroup $ \ne G$ is either a stabilizer of a cut in $ \mathcal{C}$, or else is minimal and is the intersection of a tower of such stabilizers. $ L(\mathcal{C}) = \cap \{ {G_\Delta }\vert\Delta \epsilon \mathcal{C}\}$ is the distributive radical of $G$, so that $G$ acts faithfully (and completely) on $\mathcal{C}$ iff $G$ is completely distributive. Every closed $ l$-ideal of $G$ is $ L(\mathcal{D})$ for some system $ \mathcal{D}$. A group $ G$ in which every nontrivial $o$-block supports some $1 \ne g \epsilon G$ (e.g., a generalized ordered wreath product) fails to be complete iff $G$ has a smallest nontrivial system $\tilde \Delta$ and the restriction ${G_\Delta }\vert\Delta$ is $o$-$2$-transitive and lacks elements $\ne 1$ of bounded support. These results about permutation groups are used to show that if $H$ is an abstract $l$-group having a representing subgroup, its closed $l$-ideals form a tower under inclusion; and that if $\{ {K_\lambda }\}$ is a Holland kernel of a completely distributive abstract $l$-group $H$, then so is the set of closures $\{ K_\lambda ^ \ast \} $, so that if $ H$ has a transitive representation as a permutation group, it has a complete transitive representation.
Some results on parafree groups
Yael
Roitberg
315-339
Abstract: We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup $N$ of a parafree metabelian group $ P$ of rank $\geqslant 2$ such that $N \cdot {\gamma _2}P$ has infinite index in $ P$ is not finitely generated unless it is trivial. (2) If $x$ and $y$ are elements of a parafree group $ P$ in any variety containing the variety of all metabelian groups which are independent modulo $ {\gamma _2}P$, then the commutator $[x,y]$ is not a proper power.
$0\leq X\sp{2}\leq X$
Ralph
Gellar
341-352
Abstract: This paper studies the structure of elements $X$ satisfying $0 \leqslant {X^2} \leqslant X$ in a Dedekind $ \sigma$-complete partially ordered real linear algebra. The lollipop-shaped possible spectrum of $X$ had been described previously. Three basic example types are described, each with possible spectrum a characteristic part of the lollipop and the possibility of splitting $X$ into a sum of these types is considered. The matrix case is scrutinized. There are applications to operator theory. Contributions to the theory of convergence in partially ordered algebras are developed for technical purposes.
On a variation of the Ramsey number
Gary
Chartrand;
Seymour
Schuster
353-362
Abstract: Let $c(m,n)$ be the least integer $p$ such that, for any graph $G$ of order $p$, either $G$ has an $m$-cycle or its complement $\bar G$ has an $n$-cycle. Values of $c(m,n)$ are established for $m,n \leqslant 6$ and general formulas are proved for $ c(3,n),c(4,n)$, and $ c(5,n)$.
A new class of functions of bounded index
S. M.
Shah;
S. N.
Shah
363-377
Abstract: Entire functions of strongly bounded index have been defined and it is shown that functions of genus zero and having all negative zeros satisfying a one sided growth condition belong to this class.
Automorphisms of ${\rm GL}\sb{n}(R),\,R$ a local ring
J.
Pomfret;
B. R.
McDonald
379-388
Abstract: Let $R$ denote a commutative local ring with maximal ideal $m$ and residue field $k = R/m$. In this paper we determine the group automorphisms of the general linear group $G{L_n}(R)$ when $n \geqslant 3$ and the characteristic of $ k$ is not 2.
Simple groups of order $2\sp{a}3\sp{b}5\sp{c}7\sp{d}p$
Leo J.
Alex
389-399
Abstract: Let $ {\operatorname{PSL}}(n,q)$ denote the projective special linear group of degree $n$ over $ {\text{GF}}(q)$, the field with $q$ elements. The following theorem is proved. Theorem. Let $G$ be a simple group of order ${2^a}{3^b}{5^c}{7^d}p,a > 0,p$ an odd prime. If the index of a Sylow $p$-subgroup of $G$ in its normalizer is two, then $G$ is isomorphic to one of the groups, $ {\operatorname{PSL}}(2,5),{\operatorname{PSL}}(2,7),{\operatorname{PSL}}(2,9),... ...\operatorname{PSL}}(2,25),{\operatorname{PSL}}(2,27),{\operatorname{PSL}}(2,81)$, and $ {\operatorname{PSL}}(3,4)$.
The study of commutative semigroups with greatest group-homomorphism
Takayuki
Tamura;
Howard B.
Hamilton
401-419
Abstract: This paper characterizes commutative semigroups which admit a greatest group-homomorphism in various ways. One of the important theorems is that a commutative semigroup $ S$ has a greatest group-homomorphic image if and only if for every $a \in S$ there are $b,c \in S$ such that $abc = c$. Further the authors study a relationship between $S$ and a certain cofinal subsemigroup and discuss the structure of commutative separative semigroups which have a greatest group-homomorphic image.
Functions and integrals
J.
Malone
421-447
Abstract: In §2 a mapping of nonnegative functions is defined to be an integral if it has the following properties: $I(f) \geqslant 0,I(f) < \infty$ for some $f$, if $f \leqslant g$ then $I(f) \leqslant I(g),I(f + f) = 2I(f),I(\sum\nolimits_{n = 1}^\infty {{g_n}) \leqslant \sum\nolimits_{n = 1}^\infty {I({g_n})} }$. Given an integral $ I$ a nonnegative function $ f$ is defined to be a measurable function if $I(f + g) = I(f) + I(g)$ for all nonnegative functions $g$. If $f,g,({g_n})_{n = 1}^\infty $ are measurable functions then the following functions are measurable: $ f + g,af$ for all $a \geqslant 0,\sum\nolimits_{n = 1}^\infty {{g_n},f - g}$ if $ f - g \geqslant 0$ and $I(g) < \infty$; also $\sum\nolimits_{n = 1}^\infty {I({g_n}) = I(\sum\nolimits_{n = 1}^\infty {{g_n})} }$. An example shows if $f,g$ are measurable functions then $\max \{ f,g\}$ may fail to be a measurable function. If an integral has the property that if $ f,g$ are measurable functions then $\max \{ f,g\} $ is a measurable function, then the following functions are also measurable: $ \min \{ f,g\} ,\vert f - g\vert,\sup {g_n}$ and under certain conditions ${\lim _{n \to \infty }}\sup {g_n},\inf {g_n},{\lim _{n \to \infty }}\inf {g_n}$ whenever $ ({g_n})_{n = 1}^\infty$ is a sequence of measurable functions. A theorem similar to Lebesgue's dominated convergence theorem is shown to hold. In §1 the Lebesgue integral, which does not in general have the properties required to be an integral as defined in §2, is used to obtain an integral $ {\mathbf{U}}$ which does. If $\mu$ is an outer measure and ${\mathfrak{M}_\mu }$ is the $\sigma $-algebra of $ \mu$-measurable sets then the set of measurable functions defined in §2 for the integral $ {\mathbf{U}}$ contains the usual set of $ {\mathfrak{M}_\mu }$-measurable functions. $ {\mathbf{U}}$ has the property that if $f$ is a $ {\mathfrak{M}_\mu }$-measurable function and if $ \int_X {fd\mu }$ denotes the Lebesgue integral of $f$ on a set $X$ then $\int_X {fd\mu = {{\mathbf{U}}_X}fd\mu }$. In §3 it is shown that an integral $ I$ defined on a set $ X$ induces an outer measure $\mu$. If $\mu$ is a regular outer measure, a representation theorem holds for $I$: if $f$ is a nonnegative function and ${\mathbf{U}}$ is the integral of §1 then $I(f) = {U_X}fd\mu$. Regardless of whether or not the outer measure $\mu$ is regular a similar theorem can be obtained: if $f$ is a nonnegative ${\mathfrak{M}_\mu }$-measurable function then $ I(f) = {{\mathbf{U}}_X}fd\mu$. The relationship between $\mu $-measurable sets and measurable functions is explored.
A Radon-Niko\'ym theorem for operator-valued measures
Hugh B.
Maynard
449-463
Abstract: The purpose of this paper is to obtain a characterization of indefinite integrals of vector-valued functions with respect to countably additive operator-valued measures with finite variation. This result is then specialized to several simpler situations.
Canonical neighborhoods for topologically embedded polyhedra
Robert
Craggs
465-490
Abstract: D. R. McMillan has shown that in any neighborhood of a compact two sided surface in a $3$-manifold there is a closed neighborhood of the surface which is the sum of a solid homeomorphic to the cartesian product of the surface with the unit interval and some small disjoint cubes-with-handles each of which intersects the cartesian product in a disk on its boundary. In the present paper the author generalizes this notion of canonical neighborhood so that it applies to topological embeddings of arbitrary polyhedra in $3$-manifolds. This is done by replacing the cartesian products by small regular neighborhoods of polyhedral approximations to the topological embeddings.
On trigonometric series associated with separable, translation invariant subspaces of $L\sp{\infty }(G)$
Ron C.
Blei
491-499
Abstract: $G$ denotes a compact abelian group, and $ \Gamma$ denotes its dual. Our main result is that every non-Sidon set $E \subset \Gamma$ contains a non-Sidon set $ F$ such that $L_F^\infty (G) = { \oplus _l}1_{i = 1}^\infty {C_{{F_i}}}(G)$, where the ${F_i}$'s are finite, mutually disjoint, and $ \cup _{i = 1}^\infty {F_i} = F$.
A proof that $\mathcal{C}^2$ and $\mathcal{T}^2$ are distinct measures
Lawrence R.
Ernst
501-508
Abstract: We prove that there exists a nonempty family $X$ of subsets of $ {{\text{R}}^3}$ such that the two-dimensional Carathéodory measure of each member of $X$ is less than its two-dimensional $\mathcal{T}$ measure. Every member of $ X$ is the Cartesian product of 3 copies of a suitable Cantor type subset of ${\text{R}}$.