ABSTRACT
In this chapter, the author uses a general ideal approach and identifies two properties that an ideal may possess sufficient to allow an ideal Piccard-Pettis theorem. This general version of the theorem admits application to measure theory by which we obtain a theorem similar in flavor to the aforementioned theorem of H. Steinhaus. The author would like to thank Professor Janusz Matkowski for making us aware of several important references, and the referee for supplying Theorem which enabled us to simplify the proofs of other theorems. The referee also pointed out that stronger versions exist for some of our measure-theoretic applications.
