Is there any connection between the trace sigma-algebra and probabilistic conditioning? Suppose we have a sub sigma-algebra $\mathcal{G}$ and a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Let $A$ be an event in $\mathcal{G}$ and $E$ an event in $\mathcal{F}$.
Do we have any relationship between $\mathbb{P}(E|\mathcal{G})$ and $\mathbb{P}(E|\mathcal{G}\cap A)$, where $\mathcal{G}\cap A$ denotes the trace sigma-algebra (i.e. all the intersections of the events in $\mathcal{G}$ with $A$)? Is there any connection between the concept of trace sigma algebra and the conditional measure $\mathbb{P}(\cdot|A)$ which I will denote $\mathbb{P}_A$?
I was thinking of relationships of the type $\mathbb{P}(E|\mathcal{G})(\omega)=\mathbb{P}(E|\mathcal{G}\cap A)(\omega)$ for all $\omega \in A$ or similar to $\mathbb{P}(E|\mathcal{G}\cap A)=\mathbb{P}_A(E|\mathcal{G})$. If not, what is the concept of trace sigma-algebra used for?