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?](https://mathoverflow.net/questions/14586/when-can-a-function-be-recovered-from-a-distribution)


Can someone please clarify this for me?



Difference between measures and distributions
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>


