D. H. Fremlin

Compact operators in Banach lattices

Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.

Pointwise compact sets of measurable functions

If X is a compact Radon measure space, and A is a pointwise compact set of real-valued measurable functions on X, then A is compact for the topology of convergence in measure (Corollary 2H). Consequently, if Xo,..., Xn are Radon measure spaces, then a separately continuous real-valued function on Xo×X1×...×Xn is jointly measurable (Theorem 3E). If we seek to generalize this work, we encounter some undecidable problems (§4).

Measurable Functions and Almost Continuous Functions.

I show that if (X, μ) is a Radon measure space and Y is a metric space, then a function from X to Y is μ-measurable iff it is almost continuous (=Lusin measurable). I discuss other cases in which measurable functions are almost continuous.

A positive compact operator

I give an example of a positive compact operator on L2[0,1] which is not representable by any Lebesgue measurable function on [0,1]2. This example can be adapted to answer a question of H.H.Schaefer (§4 below).

Riesz spaces with the order-continuity property. I

I continue to investigate Riesz spaces E with the property that every positive linear map from E to an Archimedean Riesz space is sequentially order-continuous. In order to give a criterion for the product of such spaces to be another, we are forced to investigate their internal structure, and to develop an ordinal hierarchy of such spaces.

Pointwise compact and stable sets of measurable functions

In a series of papers culminating in [9], M. Talagrand, the second author, and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the theory of Pettis integration, were solved subject to various special axioms. It was left unclear just how far the special axioms were necessary. In particular, several results depended on the fact that it is consistent to suppose that every countable relatively pointwise compact set of Lebesgue measurable functions is ‘stable’ in Talagrands sense, the point being that stable sets are known to have a variety of properties not shared by all pointwise compact sets. In the present paper we present a model of set theory in which there is a countable relatively pointwise compact set of Lebesgue measurable functions which is not stable and discuss the significance of this model in relation to the original questions. A feature of our model which may be of independent interest is the following: in it, there is a closed negligible set Q ⊆ [0, 1] 2 such that whenever D ⊆ [0,1] has outer measure 1, then has inner measure 1 (see 2G below). We embark immediately on the central ideas of this paper, setting out a construction of a partially ordered set which forces a fairly technical proposition in measure theory (IS below); the relevance of this proposition to pointwise compact sets will be discussed in §2.

On partitions of the real line

Answering a question of Sierpinski, we prove that the real line is not necessarily the disjoint union of {btℵ}1 non-emptyGσ sets.

Uncountable powers of\(\underset{\raise0.3em\hbox{

AbstractMoran ([6]) and, independently, Kemperman & Maharam ([4]) have shown that

}}{R}\) can be almost Lindelöf

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