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

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SUMMARY

Every compact subset K of R can serve as the support for a Borel measure m, where m(R) equals m(K) and is finite. Additionally, any proper compact subset L of K will have a measure strictly less than m(K). The existence of such a measure is established through the use of a dense countable subset of K, which facilitates the construction of the Borel measure.

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

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