Here's another interesting example.

Suppose you're going to observe Y in order to infer some parameter X. You know that P(x=c | y) = 1/2^(c-y).

• You set your prior to P(x=c) = 1 for all c. Very improper.
• You make an observation, y=1.
• You update: P(x=c) = 1/2^(c-1)
• You can now normalize P(x) so its area under the curve is 1.
• You could have done that, regardless of what you observed y to be. Your posterior is guaranteed to be well formed.

You get well formed probabilities out of this process. It converges to the same result that Bayesianism does as more observations are made. The main constraint imposed is that the prior must "sufficiently disagree" in predictions about a coming observation, so that the area becomes finite in every case.

I think you can also get these improper priors by running the updating process backwards. Some posteriors are only accessible via improper priors.