**Definition.** An outer measure on a nonempty set is a function that satisfies

1) ,

2) if $A\subset B$,

3) .

The most common way to obtain outer measure is to start with a family of “elementary sets” on which a notion of measure is defined (such as rectangles in the plane) and then to approximate arbitrary sets “from the outside” by countable unions of members of . The precise construction is as follows.

**Proposition.** Let and be such that , and . For any , define

and .

Then is an outer measure.

Proof. For any , such that (take for all ) so the definition of makes sense. Obviously (take for all ), and for because the set over which the infimum is taken in the definition of includes the corresponding set in the definition of . To prove countable subadditivity, suppose and . For each there exists such that and . But then if , we have and , whence . Since is arbitrary, we are done.

**Definition.** If is an outer measure on , a set is called -measurable if

for all .

The inequality holds for any and , so to prove that is -measurable, it suffices to prove that the reverse inequality. The latter is trivial if , so we see that is -measurable iff

for all such that .

If is a “well-behaved” set such that , the equation says that the outer measure of , is equal to the “inner measure” of , .

**Caratheodory’s Theorem.** If is an outer measure on , the collection of -measurable sets is a -algebra, and the restriction of to is a complete measure.

Proof. First, we observe that is closed under complements since the definition of -measurability of is symmetric in and . Next, if and ,

.

But , so by subadditivity,

,

and hence

.

It follows that , so is an algebra. Moreover, if and ,

,

so is finitely additive on .

To show that is a -algebra, it will suffice to show that is closed under countable disjoint unions. If is a sequence of disjoint sets in , let and . Then for any ,

,

so a simple induction shows that . Therefore,

,

and letting we obtain

.

All the inequalities in the last calculation are thus equalities. It follows that and taking that , so is countably additive on . Finally, if , for any we have

,

so that . Therefore is a complete measure.

Let go of all thoughts of limitations about how you should or should not behave based on how old or young you are. Age is nothing but a number, “an issue of mind over matter. If you don’t mind, it doesn’t matter.” ~ Mark Twain

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 28-30.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.