I'm interested in seeing (or generating) lots of examples of measures $\mu$ on $\mathbb{R}$ such that $\mu \sim \mu * \mu$. I'd love a reference (or even just a name) for these measures or, alternatively, a way to make up examples. I'd like to at least know that there are many measure classes whose representatives satisfy $\mu \sim \mu * \mu$, if we can't explicitly generate a family of examples.
At first, I thought it might be easier to think about these measures through the Fourier transform $\hat{\mu}$, since the Fourier transform of $\mu * \mu$ is just the product of $\hat{\mu}$ with itself, but I don't know how to determine equivalence of the measures $\mu$ and $\mu * \mu$ in terms of their Fourier transforms.
In my actual application, I also want the Fourier transform to decay at infinity, i.e. $\lim_{|t|\rightarrow \infty} \hat{\mu}(t) = 0$, so I would also appreciate comments/references about generating those examples in particular, but I'm interested in any examples where $\mu \sim \mu * \mu$, whether they meet my decay condition or not. If it's not easy to generate examples, I'd still like to see whether there's a "large" set of measure classes with these properties.