Countable network weight and multiplication of Borel sets

A space Borel multiplies with a space if each Borel set of is a member of the -algebra in generated by Borel rectangles. We show that a regular space Borel multiplies with every regular space if and only if has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for to Borel multiply with every metric space.

     

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-emptyG _σ sets.     

Embedding Spaces of Finite Length in R3

I refine a theorem from [3] to show that if (X, ) is any metricspace of finite length, it can be embedded in a compact connectedsubset of R³ of finite length in such a way as to preserve themeasure µ     

Symmetry and measurability

In the Hewitt–Savage 0-1 law, symmetric measurable sets are considered in a countable product of a σ-algebra Σ with itself. Therefore, it may be of interest to find simple generators for these sets in terms of the generators of Σ. We put the problem into a more general framework of action groups and as an application find simple generators for the finite product case. We also have a partial result in the countable product case.     

The use of body hydrogen as an internal standard in the measurement of nitrogen in vivo by prompt neutron capture gamma-ray analysis

The previously established technique of analysis of neutron capture γ-rays for the determination of nitrogen in-vivo has been modified to make estimates of absolute quantities of nitrogen mass. Hydrogen is used as an internal standard and the nitrogen mass is determined from the observed ratio of nitrogen to hydrogen counts, together with the hydrogen mass, estimated from body parameters. Phantom and cadaver studies indicate that nitrogen mass can be estimated to ±4% or better by this means.     

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 X_o,..., X_n are Radon measure spaces, then a separately continuous real-valued function on X_o×X₁×...×X_n is jointly measurable (Theorem 3E). If we seek to generalize this work, we encounter some undecidable problems (§4).