The Exterior Measure

Exterior Measure

The notion of exterior measure is the first of two important concepts needed to develop a theory of measure. Loosely speaking, the exterior measure assigns to any subset of a first notion of size.

We shall first define the rectangle and cube in .

Rectangle & Cube

A (closed) rectangle in is given by the product of one-dimensional closed and bounded intervals Moreover, a cube is a rectangle for which for all .

Theorem

Every open subset of with can be written as a countable union of almost disjoint closed cubes.

Exterior Measure

If is any subset of , then the exterior (outer) measure of is a function that where the infimum is taken over all countable coverings of closed cubes.

Proposition

Follow from the definition, we immediately have

• .
• The exterior measure of a rectangle is its volume.
• .
• is translation invariant.

Proof A single point set can be viewed as a cube of side length zero.

Properties of Exterior Measure

Monotonicity

If , then .

Proof Any countable covering of also covers .

Corollary

Any bounded has finite exterior measure.

Proof If is bounded, then for some .

Countable Subadditivity

If , then .

Proof For each , we can find a countable covering of closed cubes such that as is the infimum. Note that covers . So we have As this holds for all , taking the limit gives the desired inequality.

Open Sets Approximation

If , then where is open.

Proof By monotonicity, it is clear that . For the other direction, we can find a countable covering of closed cubes such that . Then, we expand each to an open cube containing , and such that . Then let which is open, and by countable subadditivity, we have Since is arbitrary, we have the desired inequality.

Proposition

The exterior measure is also the infimum size of rectangular coverings:

Proposition

If , and , then

Remark

One cannot conclude in general that if is a disjoint union of subsets of , then In fact it holds when the sets are not highly irregular or “pathological” but are measurable.

Proposition

If a set is the countable union of almost disjoint cubes , then where cubes are almost disjoint if their interiors are disjoint: for all .