almost equality

In measure theory, two functions $f, g\colon X \to Y$ are **almost-everywhere equal**, or **almost equal**, if their equaliser

$\{ a\colon X \;|\; f(a) = g(a) \}$

is a full set. We are usually only interested when the functions $f$ and $g$ are measurable, although this is not technically necessary. We also may consider when $f$ and $g$ are almost functions (partial functions whose domains are full).

As we need to know what a full set is, there is no notion of almost equality on an arbitrary classical measurable space. However, if the measurable space is equipped with such a notion (as is always the case with a Cheng measurable space or a localisable measurable space), then we have almost functions. Of course, a measure space also has plenty of structure for this. The morphisms between measurable locales are also inherently considered only up to almost equality.

Besides measure theory, the concept applies whenever we have a notion of something being true (in this case, that two functions are equal) almost everywhere.

Created on July 30, 2012 12:18:59
by Toby Bartels
(98.16.162.107)