Is $\Omega^\infty (g(n)$ a more general version of $\Omega(g(n))$?
The phrase "for infinitely many integers" means "for an infinite set of integers". This strange definition is what you get when you negate little $o$. That is, for any two functions $f,g \geq 0$, either $f = o(g)$ or $f = \Omega^\infty(g)$. In some cases $f = \Omega^\infty(g)$ is enough to derive certain consequences, and so is a useful property.