The responses to [another question](https://mathoverflow.net/questions/17732/) clarifies that the best known examples of distributions that are not measures, are the derivatives of the delta and such. What I want to know is: Is that the only way a distribution is not a measure?



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> Are there distributions that are not measures, their derivatives, and their derivatives, and so on?
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Distributions more complicated than the Dirac δ and derivatives
There is a structure theorem for distributions that shows that they are all (possibly infinite, but locally finite) sums of derivatives. 


Here's a proper statement. Let $T$ be a distribution on $\mathbb{R}^n$. Then there exists continuous functions $f\_{\alpha}$ such that $T = \sum\_{\alpha} (\frac{\partial}{\partial x})^{\alpha} f\_{\alpha}$, where for each bounded open set $\Omega$, all but a finite number of the distributions $(\frac{\partial}{\partial x})^{\alpha} f\_{\alpha}$ vanish identically on $\Omega$. Here $\alpha$ is a multiindex.


I recommend perusing the beautiful little book "A Guide to Distribution Theory and Fourier Transforms" by Strichartz. The above theorem is discussed in section 6.2.


