Your new first paragraph is not the definition. Partly it goes opposite the definition and partly it is orthogonal. It is so confused, I’m surprised that the other material is (or looked) correct. You should separate your consideration of continuous priors from improper priors. An example of an improper prior in a discrete setting is the uniform prior on positive integers. Another example is the prior p(n) = 1/n.
Edited to add this.
Your new first paragraph is not the definition. Partly it goes opposite the definition and partly it is orthogonal. It is so confused, I’m surprised that the other material is (or looked) correct. You should separate your consideration of continuous priors from improper priors. An example of an improper prior in a discrete setting is the uniform prior on positive integers. Another example is the prior p(n) = 1/n.
I am also confused. More specifically, improper priors are priors that integrate to infinity and thus cannot be normalized.
That’s almost the definition, except that improper priors are not priors.
Is that your confusion?
No, I mean I share your confusion that the rest of the conversation appeared reasonable given the incorrect definition in the post.
Sorry. Probably part of the miscommunication is that I used “confused” to describe Daniel LC and “surprised” to describe myself.