Difference between measures and distributions On the one hand, Wikipedia suggests that every distribution defines a Radon measure:


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*http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions
On the other hand, Terry Tao and LK suggest not:


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*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?
 A: 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
A: Do you mean this sentence:

Conversely, essentially by the Riesz representation theorem, every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.

The condition that the distribution be non-negative for non-negative functions is non-trivial. Not every distribution satisfies this, so not every distribution is a Radon measure.
The fundamental examples are the delta function at a point (which is a measure) and its derivatives (which are not measures).
A: I think the decisive point is continuity with respect to different topologies.
Let $C$ be the space of continuous functions of compact support and $D$ the space of smooth functions of compact support. 
The inclusion $D\hookrightarrow C$ is a continuous map when you give both spaces the corresponding inductive limit topology.
That means, that every continuous linear functional of $C$, i.e., each Radon-measure, defines a continuous linear functional on $D$, i.e., a distribution. But not every distribution extends to a continuous linear map on $C$. Examples are the derivatives of the Dirac distribution.
The line in Wikipedia relates to an important property of linear functionals on $C$: if such a functional is positive, i.e., if it maps functions $f\ge 0$ to numbers $\ge 0$, then it is AUTOMATICALLY CONTINUOUS.
This is a very important fact, though it is not hard to prove.
A: Measures are dual to continuous functions, whereas distributions are derivatives of them.
