If you do not have a limit, but want to talk about it you say "ultralimit".

Let $(K_n)$ be your sequence of compact sets (in any complete metric space not nesessury $L^2(\Omega)$). Consider sequence of functions $$f_n=\mathop{\rm dist}\nolimits_{K_n}.$$ It is a a sequence of 1-Lipschtz functions, so you can pass to its ultralimit $f_\omega$ for a fixed in advance ultrafilter $\omega$. The zero set $K_\omega$ of $f_\omega$ can be considered as the ultralimit of $K_n$.

If you do not have a limit, but want to talk about it you say ultralimit.

Let $(K_n)$ be your sequence of compact sets (in any complete metric space not nesessury $L^2(\Omega)$). Consider sequence of functions $$f_n=\mathop{\rm dist}\nolimits_{K_n}.$$ It is a sequence of 1-Lipschtz functions, so you can pass to its ultralimit $f_\omega$ for a fixed in advance ultrafilter $\omega$. The zero set $K_\omega$ of $f_\omega$ can be considered as the ultralimit of $K_n$.