I have a vague understanding of (sub)sigma-algebras in probability theory representing information, especially when conditioning. I want to make this intuition more exact. Suppose we are working in a probability space $(\Omega, F, P)$. I understand that $\{\emptyset,\Omega\}$ represents no information and that $F$ represents all "possible" information. For some specific other examples I also have intuition on their meaning. For a random variable $X$, $\sigma(X)$ represents knowing the value of $X$. For an event $A\in F$, $\sigma(1_A)=\{\emptyset, \Omega, A, A^c\}$ represents knowing whether $A$ happened or not.
Now I'm interested in how to interpret conditioning on some arbitrary (usually infinite) subsigma-algebra $G\subset F,$ but the intuition from the above examples seems to break down. I first suspected that we could interpret conditioning on $G$ as knowing for each event in $G$ whether it occurs or not, but this leads to problems. If we take for example $F$ to be the Borel sigma-algebra on $\mathbb{R}$, then this would mean that any sigma-algebra containing all singletons (which are often null sets) leads to "full knowledge". Since we could then for each point $x\in\mathbb{R}$ determine whether the event $\{x\}$ occurred, which should allow us to know exactly in which point of $\mathbb{R}$ we are. In particular, the sigma-algebra $H:=\sigma(\{x\in\mathbb{R}\})$ would contain all the information that the full Borel sigma-algebra $F$ contains. It seems to me that this cannot be the case, so I'm wondering how to solve this flaw in my understanding. What information does conditioning on $F$ give that $H$ does not?
I suspect that my intuition breaks down because sigma-algebras in principle only allow countable operations, but to know in what point $x\in\mathbb{R}$ I am, I need to check uncountably many singletons. Still, I'm having trouble formulating what the events in a conditioning sigma-algebra then represent in terms of information. It seems to be something like "you know of each event individually whether it occurs, but you can't combine this information for different events", which seems strange/unintuitive to me.