Suppose is a metric space, is a families of subsets of , and is a function on such that

we can obtain a measure through a prodecure called Caratheodory’s construction. Define

for any set . Then

This means is monotone, then it is reasonable to define

and are (outer) measures. Moreover

This means all open set are measurable in . If contains Borel sets of , then is Borel regular measure.

Suppose , is the Borel set of ,

Then the Caratheodory’s construction will yield the dimensional Hausdorff measure.

**Lemma:** is a (separable?) metric space, is a measure on , , is measurable whenever is a Borel subset of , define

is obtained from the Caratheodory’s construction from on all Borel subsets of . Then

where . *Proof:* Find Borel partitions of such that is a refinement of , or all subsets in are unions of that of . And also assume

Let be the characteristic function of , is from partition .

From Levy’s monotone convergence thm,

by the defintion of .

Note that the defition of implies for any , hence for very Borel set . Since all Borel set are measurable, for any partition

combining with the result before,

**Theorem:** Suppose is a Lipschitzian map. is nonegative number, then

whenever is a Borel subset. *Proof:* Let us apply the above lemma with on , then

For any Borel set , any covering of by sets diameter yields a covering of of diameter , then

By taking the limit, .

**Corollary: **For any connected set

*Proof:* Since is a regular measure on , we can find Borel set containing with the same measure. Choose , define , . Then is Lipschitzian map and . By the previous theorem