标题： 点态紧致且稳定的可测函数集 显示英文标题

标题： Pointwise compact and stable sets of measurable functions

摘要： 在一系列论文中，第二作者M.塔拉格兰德及其他研究者详细探讨了点态紧的可测函数集的性质与结构。一些本身有趣且对佩蒂斯积分理论重要的问题，在满足各种特殊公理的情况下得到了解决。然而，这些特殊公理究竟有多大的必要性仍不清楚。 特别是，几个结果依赖于这样一个事实：假定每个关于勒贝格可测函数的可数相对点态紧集在塔拉格兰德的意义上是“稳定的”是一致的；关键在于，稳定的集合已知具有一些并非所有点态紧集都具有的特性。 在本文中，我们构建了一个集合论模型，在该模型中存在一个关于勒贝格可测函数的可数相对点态紧集，它不是稳定的，并讨论了这个模型对于原始问题的意义。我们的模型的一个可能具有独立兴趣的特点如下：在这个模型中，存在一个闭的零测集\( Q \subseteq [0,1]^2 \)，使得当\( D \subseteq [0,1] \)具有外测度1时，集合\( Q^\{{-1}\}[D]={x: (∃y ∈ D)((x, y) ∈ Q)}\)具有内测度1。 显示英文摘要

摘要： In a series of papers, 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 Talagrand's 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 subseteq [0,1]^2 such that whenever D subseteq [0,1] has outer measure 1 then the set Q^{-1}[D]= {x:(exists y in D)((x,y) in Q)} has inner measure 1.