Mathematics Colloquium


Department of Mathematics
University of Nebraska at Omaha


WHEN:
On Friday, December 05, 2003 at 2:30 PM

WHERE:
Durham Science Center, Room 255

WHAT:


Andrzej Roslanowski

University of Nebraska at Omaha


will give a talk on

Results that made me happy


ABSTRACT:

We are going to deal with two, as it occurs closely related, problems concerning real functions. The first one is the question if it is possible that all superposition-measurable functions are measurable.


Definition:    A function $ f:{\mathbb{R}}^2\longrightarrow{\mathbb{R}}$ is superposition-measurable (in short: sup-measurable) if for every Lebesgue measurable function $ g:{\mathbb{R}}\longrightarrow{\mathbb{R}}$ the superposition $ f_g:{\mathbb{R}}\longrightarrow{\mathbb{R}}:x\mapsto f(x,g(x))$ is Lebesgue measurable.


The interest in sup-measurable functions comes from differential equations and the question for which functions $ f:{\mathbb{R}}^2\longrightarrow{\mathbb{R}}$ the Cauchy problem $ y'=f(x,y)$, $ y(x_0)=y_0$ has a unique almost-everywhere solution in the class $ AC_l({\mathbb{R}})$ of locally absolutely continuous functions on $ {\mathbb{R}}$. Grande and Lipinski proved in 1978 that, under CH, there is a non-measurable function which is sup-measurable. Later the assumption of CH was weakened, however the question if one can build a non-measurable sup-measurable function in ZFC remained open.

In a (still unpublished) paper with Saharon Shelah, we answered this question. We proved that, consistently, every sup-measurable function is Lebesgue measurable.

The second problem we deal with is von Weizsäcker problem concerning a generalization of Blumberg theorem (which states that every real function (on $ {\mathbb{R}}$) is continuous when restricted to a dense subset of $ {\mathbb{R}}$). In the same joint paper with Shelah we proved that, consistently, every real function is continuous when restricted to a set of positive outer measure.

Though, in the talk, I will not present how the two consistency results are obtained, I will show how / why the two questions are related, and what are the backgrounds and motivation for them.


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Last modified:   Thu Nov 20 17:46:15 CST 2003