T.J. Kaczynski: Boundary functions
Boundary Functions
T.J. Kaczynski
Doctoral dissertation abstract, University of Michigan
Let H denote the set of all points in the Euclidean plane having positive
y-coordinate, and let X denote the x-axis. If p is a point of X, then by an
arc at p we mean a simple arc v, having one endpoint at p, such that
v - {p} ( H. Let f be a function mapping H into the Riemann sphere. By a
boundary function for f we mean a function t defined on a set E ( X such
that for each p ( E there exists an arc v at p for which
lim f(z) = t(p).
z -> p
z ( v
The set of curvilinear convergence of f is the largest set on which a
boundary function for f can be defined; in other words, it is the set of all
points p ( X such that there exists an arc at p along which f approaches a
limit. A theorem of J.E. McMillan states that if f is a continuous function
mapping H into the Riemann sphere, then the set of curvilinear convergence
of F is of type F(sd). In the first of two chapters of this dissertation
we give a more direct proof of this result than McMillan's, and we prove,
conversely, that if A is a set of type F(sd) in X, then there exists a
bounded continuous complex-valued function in H having A as its set of
curvilinear convergence. Next, we prove that a boundary function for a
continuous function can always be made into a function of Baire class 1
by changing its values on a countable set of points. Conversely, we show that
if t is a function mapping a set E ( X into the Riemann sphere, and if t
can be made into a function of Baire class 1 by changing its values on a
countable set, then there exists a continuous function in H having t as a
boundary function. (This is a slight generalization of a theorem of
Bagemihl and Piranian.) In the second chapter we prove that a boundary
function for a function of Baire class e > 1 in H is of Baire class at
most e + 1. It follows from this that a boundary function for a
Borel-measurable function is always Borel-measurable, but we show that
a boundary function for a Lebesgue-measurable function need not be
Lebesgue-measurable. The dissertation concludes with a list of problems
remaining to be solved.
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