Show that we get an equivalent notion if in the above...

Show that we get an equivalent notion if in the above definition we replace the word 'intervals' by any of these: 'open intervals', 'closed intervals', 'the intervals of the form $(a, b]$, 'the intervals of the form $[a, b)$ '.

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Key Concepts

Interval Coverings

In the context of measure theory, a common strategy is to approximate the size of a set by covering it with intervals. The key idea is that various families of intervals—open, closed, or half-open—can be used because they can approximate each other arbitrarily closely. This flexibility allows one to replace the general family of intervals with any specific type without affecting the resulting measure.

Equivalent Definitions

Even though different types of intervals possess distinct topological properties, using them in the definition of measures like the outer measure leads to equivalent notions. This is because each family of intervals is sufficiently 'rich' to cover sets in a way that their infima over coverings agree, highlighting the robustness and consistency of the measure-theoretic framework.

Outer Measure

An outer measure is a function defined on all subsets of a space that assigns a non?negative extended real number to a set, typically defined as the infimum of the total 'size' (like length) over all collections of simple sets (such as intervals) covering the set. It forms the basis for constructing more refined measures by ensuring countable subadditivity and providing an upper bound on the 'size' of any subset.

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