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.