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For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work with in general. It's always easy to re-normalize at the end, if needed.

I cannot for the life of me remember what these more general objects are called, though. Are they just "distributions" or is there a more specific name? Again, the name for the members of a function space with all the properties of a PDF except the requirement that they integrate to 1.

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Are you thinking of a measure? – VSJ Sep 13 '12 at 19:46
Well, a measure with finite total mass. – Deane Yang Sep 14 '12 at 2:44

The name is "kernel of a probability distribution" or "unnormalized kernel". This usage seems to be mostly prevalent in Bayesian Statistics as explained here

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For me as a probabilist this terminology sounds strange (and I did not know it). In probability a kernel would be more something like a (not necessarily normalized) Markov kernel and we would say ``finite measure''. – Wolfgang Loehr Sep 14 '12 at 9:54
Hi Wolfgang. I agree with you that the terminology is quite unfortunate, especially since the word kernel could correspond to so many different things. As far as I can observe, the terminology Markov kernel is very widely used in statistics as well. However, I think that "finite measure" would be an abuse of terminology in that setting. – an12 Sep 14 '12 at 11:37

Measures ? They are also more restrictive subsets of measure like finite measures or $\sigma$-finite measures.

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I don't think the question is about measures per se, but about their Radon-Nikodym derivatives with respect to their dominating measure, whatever that is. For instance, the PDF function integrating to 1 in the question can be thought of as the Radon-Nikodym derivative of some probability measure with respect to its dominating Lebesgue measure. – an12 Sep 14 '12 at 0:38
Only Scot Free can say what he means, and he has not chosen to return and do so. – Gerald Edgar Sep 17 '12 at 13:23
Hi Gerald. Yes, indeed, only he can say what he meant! – an12 Sep 17 '12 at 14:37

For an absolutely continuous finite Borel measure $\mu(dx)=f(x)dx$ on $\mathbb R$, if $\mu(\mathbb R)\neq 1$ then $f$ is sometimes called the "intensity" of the measure.

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Mathematicians call $f$ the "density" or the "Radon-Nikodym derivative" of the measure. – Gerald Edgar Sep 17 '12 at 13:22
Not the mathematicians which work with determinantal point processes. The distinction between "density" and "intensity" is often made when one speaks about the Radon-Nikodym derivative of a non-probability measure. – Adrien Hardy Oct 15 '12 at 16:38

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