Length Common Notions

Why

We want to define the length of a subset of real numbers.

Notions

We take two common notions:

1. The length of a whole is the sum of the lengths of its parts; the additivity principle.
2. The length of a whole is the at least the length of any whole it contains the containment principle.

The task is to make precise the use of “whole,” “parts,” and “contains.” We start with intervals.

Definition

By whole we mean set. By part we mean an element of a partition; in other words, a subset. By contains we mean set inclusion.

The length of an interval is the difference of its endpoints: the larger minus the smaller (see Interval Length). Two intervals are non-overlapping if their intersection is a single point or empty. The length of the union of two non-overlapping intervals is the sum of their lengths.