# finite measure which takes given closed set as its supprot.

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?