Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property:
its integral $\int\_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int\_{-\infty}^\infty F(x)\,dx=1$? It definitely can be seen as a limit of smooth functions, and its Fourier transform will be a finction, equal to $1$ at $0$ and otherwise $0$.


I think such object for sure should be considered and named by someone.



Anti-delta function?
There's quite a bit on this in Harold Jeffreys's unusual book *Theory of Probability*, in which such a function is one of the more prominent examples of “improper priors.” In [Robert, Chopin, and Rousseau – Harold Jeffreys's Theory of Probability Revisited](https://arxiv.org/abs/0804.3173) we find this:



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> For a 21st century reader, Jeffreys’s *Theory of Probability* is nonetheless puzzling for its lack of formalism, including its difficulties in handling improper priors, its reliance on intuition, its long debate about the nature of probability, and its repeated attempts at philosophical justifications. The title itself is misleading in that there is absolutely no exposition of the mathematical bases of probability theory in the sense of Billingsley (1986) or Feller (1970): “Theory of Inverse Probability” would have been more accurate. In other words, the style of the book appears to be both verbose and often vague in its mathematical foundations for a modern reader.
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Consider the upper half-plane $\{(m,s): -\infty<m<+\infty,\, 0<s<+\infty\}$ with the measure $dm\,ds/s.$ Pretend that when multiplied by some infinitely small constant (“constant” = not depending on $m$ or $s$) this becomes a probability measure on the upper half-plane. Now suppose that the conditional probability distribution of $X\_1,\dotsc,X\_n$ given $m$ and $s$ is that they are independent and normally distributed with expected value $m$ and variance $s$ (variance, not standard deviation). What then is the conditional distribution of $(m,s)$ given $X\_1,\dotsc,X\_n$? It is a perfectly ordinary probability distribution. Jeffreys proposes that that is the conditional probability distribution of the expectation and variance of a population from which $X\_1,\dotsc,X\_n$ is a random sample, when one has no other information about their values than what one gets from that sample.


Another example is the measure $c\,dp/\bigl(p(1-p)\bigr)$ whose integral over $(0,1)$ is $1$. Suppose $X\_1,\dotsc,X\_n$ are conditionally independent given $p$ and $\Pr(X\_k=1\mid p) = p$ and $\Pr(X\_k=0\mid p)=1-p$ for $k=1,\dotsc,n$. Then the conditional distribution of $p$ given $X\_1,\dotsc,X\_n$ is $\text{constant}\times p^{X\_1+\dotsb+X\_n-1}(1-p)^{n-(X\_1+\dotsb+X\_n)-1} \, dp,$ which is a perfectly ordinary probability distribution except when $X\_1+\dotsb+X\_n\in\{0,n\}$.


I think Jeffreys's doctorate was in mathematics, and he was a professor of astronomy for some time, and spent a lot of time on seismology and earth sciences, and in the '50s claimed to be able to tell the difference between seismic waves resulting from nuclear tests and those from earthquakes when the U.S. government was denying that that could be done.


The physicist Edwin Jaynes defended improper priors in a number of his writings.


