TITLE: Difference between measures and distributions
QUESTION [19 upvotes]: On the one hand, Wikipedia suggests that every distribution defines a Radon measure:

http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions

On the other hand, Terry Tao and LK suggest not:

http://www.math.ucla.edu/~tao/preprints/distribution.pdf
When can a function be recovered from a distribution?

Can someone please clarify this for me?

REPLY [3 votes]: This is a summary of what I've learned about this question based on the answers of the other commenters.
[*] Any positive distribution defines a positive Radon measure.
I had naively assumed a result for distributions like The Hahn Decomposition Theorem[1] for measures, i.e. I assumed that a distribution could be expressed as the difference of two positive distributions.  If it could be, then applying Theorem [*] would yield the result that any distribution is a signed measure.
However, this is not the case.  The derivative of the delta function, i.e. δ', satisfies
δ'(f) = -f'(0).  This is not a measure.  I can't find any way of proving it's not the difference of two positive distributions, other than by contradiction using the above result.
[1] http://en.wikipedia.org/wiki/Hahn_decomposition_theorem