## Lecture 3: The outer measure generated by a measure

I’ll post lecture 1 and 2 soon.

In lecture 1 and 2, I defined a measure on a semiring and an outer measure on the power set of . Our goal is that we want to define a measurable set As we recall, a measurable set satifies the following condition: for any Notice that here is an outer measure.

In this lecture, a measure is extended to an outer measure As proved before, the collection of measurable sets with respect to this outer measure is a We will show that Thus, semirings are a basic collections of sets which a measure theory can be built on.

Let a measure space be fixed. The outer measure is defined as follows:

Theorem:is an outer measure.

**Proof: **

i) Let for Then Thus

ii) Let and We consider a sequence of sets in such that Then clearly, Thus, Further, we have This follows that Hence, is monotone.

iii) Let be a sequence of pairwise disjoint sets in If . Then clearly,

Thus we can assume Since each there exists a sequence of sets in such that and Hence, Furthermore,

we have This proves the -subadditivity of

The next theorem will show that is indeed an extension of from to

Theorem:Ifthen

*Proof:*

Let and for Then Thus,

Consider a sequence of sets in satisfying Since is a measure on semiring It is -subadditive on Therefore, we have Thus,

The outer measure generated by a measure have some nice characteristics which can be shown by the following theorem.

Theorem:Let The followings are equivalent:(1)

is -measurable.(2)

with any and(3)

with any and(4)

with any

**Proof:**