June 25, 2018

Set Theory and Analysis Program

Saharon Shelah, Hebrew University and Rutgers University
October 1st - 3rd, 2002 at 3:30 pm

Adding measures to normed forcing

Normed forcing first established its usefulness in Saharon Shelah's seminal proof that the splitting number can be greater than the unboundedness number. The essential idea was to use measures (often on finite sets) to control the splitting nodes of trees used as forcing conditions. These ideas later evolved into a sophisticated theory of "creature forcing" which is described in a short book by A. Roslanowski and S. Shelah. Lately, in more joint work with A. Roslanowski, a simple form of integration was added to the mix and has allowed the solution of several longstanding problems in the theory of measurable functions.

For example, it has been shown to be consistent with set theory that all real valued functions are continuous on a non-measurable set of reals. The same techniques have also been applied to questions dealing with superposition measurable functions; namely those function from the plane to the reals whose restriction to the graph of any measurable function is measurable. The study of a certain class of solutions to the Cauchy problem leads to the question of whether every superposition measurable function is actually measurable. Measured creatures can be used to show that the answer is consistently positive.

The first lecture will provide an introduction and overview of the main results, while the last two lectures will provide details for experts in the field.

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