Quebec Mathematical Sciences Colloquium

March 4, 2011 from 16:00 to 18:00 (Montreal/Miami time) On location

Some random thoughts about Cauchy's functional equation

Colloquium presented by Dan Stroock

In this lecture I will discuss a variation on Cauchy's functional equation (*) $f(x + y) = f(x) + f(y)$ for all $(x; y) \in R^2$: After reviewing the familiar fact that any measurable f which satises (*) must be linear, I will investigate what can be said when (*) is replaced by (**) $f(x + y) = f(x) + f(y)$ for Lebesgue almost every $(x; y) \in R^2$: Borrowing ideas from probability theory, I will show that any measurable solution to (**) is Lebesgue almost everywhere equal to a linear function. If time permits, I will also show how the same ideas apply in more exotic settings.


UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., room SH-3420