Let two (countably-generated) $\sigma-$algebras $\mathscr{F,G}$ on the event space $\mathbb{R}$ be given. I believe we also need the atoms of $\mathscr{F,G}$ to be the points of $\mathbb{R}$.
Let $\mu$ be a probability measure on the measurable space $(\mathbb{R},\mathscr{F})$ which is absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$- its density is denoted $f(x)$; define $\mu^{\mathscr{G}}$ to be the restriction of $\mu$ to the sub-measure space $(\mathbb{R}, \mathscr{G})$, and define $\mu^{\mathscr{F}}$ to be the original $\mu$ defined on $(\mathbb{R},\mathscr{F})$.
The (differential) entropy of the countably-generated $\sigma-$algebra $\mathscr{F}$ is equal to or defined to be: $$ H_{\mu}(\mathscr{F}) = \int_{\mathbb{R}} -\log[f(x)]\mathrm{d}\mu^{\mathscr{F}}x $$ and that of $\mathscr{G}$ can be defined to be: $$H_{\mu}(\mathscr{G})=\int_{\mathbb{R}}-\log[f(x)]\mathrm{d}\mu^{\mathscr{G}}x $$ To the best of my knowledge, one has that $$\mathscr{G} \subset \mathscr{F} \implies H_{\mu}(\mathscr{G}) \le H_{\mu}(\mathscr{F}) \tag{1}$$ However, this inequality is really only meaningful when $H_{\mu}(\mathscr{G})$ and $H_{\mu}(\mathscr{F})$ are finite.
Question: When are $H_{\mu}(\mathscr{G})$ and $H_{\mu}(\mathscr{F})$ finite?
I have not been able to find much information about the entropy of a sigma-algebra (w.r.t. a given probability measure of course) on the internet, and the four sources [1][2][3][4] which I have been able to find all seem to lack references which discuss the concept clearly.
Note: I am interested in infinite $\sigma$-algebras because of their use in stochastic processes. In particular, I want to be able to say that "larger $\sigma-$algebras contain more information" and to have something like $(1)$ make this statement be true in a rigorous sense. In other words I would like to be able to show that the entropy of a filtration increases with time.
