Question

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
• http://mathoverflow.net/questions/14586/when-can-a-function-be-recovered-from-a-distribution

Can someone please clarify this for me?

Answer 1

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).

Answer 2

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.

Answer 3

Measures are dual to continuous functions, whereas distributions are derivatives of them.

Answer 4

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