**Definition.** Let be a set equipped with a -algebra . A measure on (or on , or simply on if such that

1). .

2). (Countable additivity) if is a sequence of disjoint sets in , then .

2)’. if are disjoint sets in , then ,

because one can take for . A function that satisfies 1) and 2)’ but not necessarily 2) is called a finitely additive measure.

If is a set and is a -algebra, is called a measurable space and the sets in are called measurable sets. If is a measure on , then is called a measure space.

Let be a measure space. If (which implies that for all since , is called finite. If where and for all , the set is said to be -finite for . More generally, if where and for all , the set is said to be -finite for . If for each with there exists with and , is called semifinite.

**Remark. **Every -finite measure is semifinite, but not conversely.

**Examples.** 1) Let be any nonempty set, , and any function from to . Then determines a measure on by the formula . is semifinite iff for every , and is -finite iff is semifinite and is countable.

a. If for all , is called counting measure;

b. If for some , is defined by and for , is called the point mass or Dirac measure at .

2) Let be an uncountable set, and let be the -algebra of countable or co-countable sets. The function on defined by if is countable and if is co-countable is easily seen to be a measure.

3) Let be an infinite set and . Define if is finite, if is infinite. Then is a finitely additive measure but not a measure.

**Theorem.** Let be a measure space.

a). (Monotonicity) If and , then $\mu(E)\le\mu(F)$.

b). (Subadditivity) If , then .

c). (Continuity from below) If and , then .

d). (Continuity from above) If , , and , then .

Proof. a) If , then .

b) Let and for . Then the are disjoint and for all . Therefore, by a)

.

c) Setting , we have

.

d) Let ; then , , and . By c) then

.

Since , we may subtract it from both sides to yield the desired result.

**Remark. ** The condition in d) could be replaced by for some , as the first ‘s can be discarded from the sequence without affecting the intersection. However, some finiteness assumption is necessary, as it can happen that for all but .

**Definition. **If is a measure space, a set such that is called a null set. By subadditivity, any countable union of null sets is a null set, a fact which we shall use frequently. If a statement about points is true except for in some every . (If more precision is needed, we shall speak of a -null set, or -almost everywhere).

If and , then by monotonicity provided that , but in general it need not be true that . A measure whose domain includes all subsets of null sets is called complete. Completeness can sometimes obviate annoying technical points, and it can always be achieved by enlarging the domain of , as follows.

**Theorem.** Suppose that is a measure space. Let and and for some . Then is a -algebra, and there is a unique extension of to a complete measure on .

Proof. Since and are closed under countable unions, so is . If where and , we can assume that (otherwise, replace and by and ). Then , so . But and , so that . Thus is a -algebra.

If as above, we set . This is well defined, since if where $F_j\subset N_j\in\mathcal N$, then $E_1\subset E_2\cup N_2$ and so , and likewise . It is easily verified that is a complete measure on , and that is the only measure on that extends .

**Remark. **The measure is called the completion of , and is called the completion of with respect to .

“Whenever you meet a ghost, don’t run away, because the ghost will capture the substance of your fear and materialize itself out of your own substance… it will take over all your own vitality… So then, whenever confronted with a ghost, walk straight into it and it will disappear.” ~ Allan Watts

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 24-27.

[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.