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Examples of distributions

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Yes - measures induce distributions, too, in the sense that the distribution maps a test function to its integral with respect to the measure. In fact, *any* positive distribution (in the sense that it maps a nonnegative test function to a nonnegative real number) is induced by a regular Borel measure.

However, by the Lebesgue decomposition theorem, you can decompose any Borel measure into a measure that’s absolutely continuous with respect to the Lebesgue measure, a discrete measure, and a singular continuous measure. Concretely, this means that (positive) distributions are a sum of (a) an integrable function (b) a countable linear combination of Dirac deltas, and (c) a singular continuous measure. I can’t give a great description of singular continuous measures unfortunately.

That being said, the Cauchy principal value is a (not necessarily positive) distribution that can be used to define singular integral operators, and as far as I’m aware, it doesn’t arise as a measure.
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You should look into the concept of order of a distribution.

Distributions associated with (locally integrable) functions are of order 0.

Dirac distributions are of order 1.

Moreover, the order is "stable under linear combinations" so linear combinaisons of functions and Dirac are of order at most 1.

So any distribution of order at least 2 will give you an example. For instance, the derivative of a Dirac (if a distribution is of order k, its derivative is of order k+1) which is very useful for application, in PDE for instance.
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> And the dirac delta function is a distribution, and is well-defined without any hand-waving, which seems like practically the whole point for talking about distributions in the first place.


While developing a context in which the Dirac delta function makes rigorous sense was an important motivation for the work on distributions by Schwartz, having that definition is not "the whole point".  You can really do things (prove new useful theorems) with distributions that were more difficult or not possible if you only have classical functions. Look up elliptic regularity theorems, derivatives of distributions (that a distribution is infinitely differentiable even it comes from a non-differentiable classical functions is one of the properties that makes distributions so nice), and the notion of fundamental solutions to a linear PDE.

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