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I want to construct a finite measure on Borel sigma algebra of $[0,1]$ which takes given closed subset $F$ of $[0,1]$ as its support.

If there is any measure $\mu$ which takes nonzero value for every nonempty set than $v(A) = \mu (A \cup F) $ gives desired measure but such 'finite' measure does not exists Could anybody give me an example of finite measure which has non-zero value for all non-empty measurable set? even though we can find infinite measure with such property, for example, counting measure.

So I'm stuck in constructing my desired measure... Could anybody give me a help?

Thanks in advance.

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For one way to do it, hint: suppose you have a countable set A, and a measure that places a positive amount of mass at each point of A.

  • How could you arrange for this measure to be finite? (Hint: you can't put the same amount of mass at every point.)

  • What is the support set of this measure?

  • How can you choose A so that the support is F? (Hint: [0,1] is second countable, ie its topology has a countable base.)

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