Negligible Set - Derived Concepts

Derived Concepts

Let X be a set, and let I be an ideal of negligible subsets of X. If p is a proposition about the elements of X, then p is true almost everywhere if the set of points where p is true is the complement of a negligible set. That is, p may not always be true, but it's false so rarely that this can be ignored for the purposes at hand.

If f and g are functions from X to the same space Y, then f and g are equivalent if they are equal almost everywhere. To make the introductory paragraph precise, then, let X be N, and let the negligible sets be the finite sets. Then f and g are sequences. If Y is a topological space, then f and g have the same limit, or both have none. (When you generalise this to a directed sets, you get the same result, but for nets.) Or, let X be a measure space, and let negligible sets be the null sets. If Y is the real line R, then either f and g have the same integral, or neither integral is defined.