Prove Existence of Borel Measure with Compact Set as Support in R

[StatusX]

Let m be a measure on the space X. I'm told that if m(X)=1, K is a compact subset of X with m(K)=1, and K has the property that any proper compact subsets of K have measure strictly less than 1, then K is called the support of m. Then I'm asked to show that every compact subset of R is the support of some Borel measure.

Does this mean that given a compact set K in R, there exists a Borel measure m with m(R)=m(K)<infinity and m(L)<m(K) for any proper compact subset L of K? If so, I'm having a hard time understanding what such a measure would look like, or how I would prove one exists. Can anyone help me here?

 

[StatusX]

Never mind, I got it (it involved a dense countable subset of K).

 

[tacman]

Yes, that is exactly what it means. To prove the existence of a Borel measure with a compact set as its support in R, we first need to define what a Borel measure is.

A Borel measure is a measure defined on the Borel sigma-algebra of a topological space. In simpler terms, it is a measure that assigns a non-negative real number to each Borel set in the space. In this case, our space is R, so we are looking for a measure defined on the Borel sets of R.

Now, let K be a compact subset of R. We want to show that there exists a Borel measure m such that m(R)=m(K)<infinity and m(L)<m(K) for any proper compact subset L of K.

To construct such a measure, we can use the outer measure defined as follows:
m*(A) = inf{∑n=1∞ |I_n| : A ⊆ ∪n=1∞ I_n, I_n are intervals}

In other words, the outer measure of a set A is the infimum of the sums of the lengths of intervals that cover A.

Now, we define a measure m on the Borel sets of R as follows:
m(A) = m*(A∩K)

In words, the measure of a Borel set A is equal to the outer measure of the intersection of A with K.

To prove that this is a Borel measure, we need to show that it satisfies the three properties of a measure:

1. Non-negativity: Since the outer measure is non-negative, m(A) = m*(A∩K) ≥ 0 for any Borel set A.

2. Countable additivity: Let {A_n} be a countable collection of disjoint Borel sets. Then,
m(∪n=1∞ A_n) = m*(∪n=1∞ A_n ∩ K) = m*(∪n=1∞ (A_n ∩ K))
= ∑n=1∞ m*(A_n ∩ K)
= ∑n=1∞ m(A_n)

3. Outer measure of an interval: It can be shown that the outer measure of an interval is its length. So