Consider the sigma-algebra generated by an event $A_0\in\Sigma$, given by $\sigma(A_0)=\{\emptyset,A_0,A_0^c,\Omega\}$.
The Conditional expectation will be $$\large{E(X|\sigma(A_0))=\frac{\mathbf{1}_{A_0}}{\mathbb{P}(A_0)}\int_{A_0}Xd\mathbb{P}+\frac{\mathbf{1}_{A^c_0}}{\mathbb{P}(A_0^c)}\int_{A_0^c}Xd\mathbb{P}}$$
Then I analogus to the example $\sigma'=\{\emptyset,\Omega\}$, $$\large{E(X|\sigma')=\frac{\mathbf{1}_{\emptyset}}{\mathbb{P}(\emptyset)}\int_{\emptyset}Xd\mathbb{P}+\frac{\mathbf{1}_{\Omega}}{\mathbb{P}(\Omega)}\int_{\Omega}Xd\mathbb{P}}=\int_{\Omega}Xd\mathbb{P}$$
This question might be not interesting at all, broadly speaking, I am just curious how scientists decide when does undefined-term can be eliminated. This is the undefined term $\frac{\mathbf{1_\emptyset}}{\mathbb{P}(\emptyset)}$ because it's denominator is also $0$. If it's denominator is not $0$, I am perfectly fine with it, since $\frac{0}{0}$ happens, I am sort of confused.
Thank you for any help and comments
