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Posting a problem like this might help me get off my arse. This is #11 / chapter 2 of Rudin's Real and Complex Analysis.

1. The problem statement, all variables and given/known data

Let m be a regular Borel measure on a compact Hausdorff space X, assume m(X) = 1. Prove that there is a compact [tex]K \subseteq X[/tex] (the support of m) such that m(K) = 1 but m(H) < 1 for every proper compact subset H of K.

Hint (given by Rudin): Let K be the intersection of all compact K_a such that m(K_a) = 1; show that every open set V which contains K also contains some K_a. Regularity of m is needed.

2. Relevant equations

A measure m is "regular" if the following two conditions hold for every measurable E:

(1) m(E) = inf{m(V): V is an open set containing E}

(2) m(E) = sup{m(K): K is a compact subset of E}

3. The attempt at a solution

,,, work in progress

Edit: "support of X" corrected to read "support of m"

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# Homework Help: The "support" of a measure

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