Conditional expectation if the events of the sigma field have probability 0 or 1.

Question

Let $(\Omega, \sigma, P)$ a probability space, Y a $\sigma|\beta$ measurable function with $E|Y|<\infty$ and $\sigma'$ any $\sigma$-field, $\sigma' \subset \sigma$. If we also have that $$P(A) \in \{0,1\} \hspace{0.6cm} \forall A \in \sigma'$$ then $$E(Y|\sigma')=E(Y) \hspace{0.5cm}\text{almost surely}$$ I'd want to proof this property but I can't see why $$\int_AEYdP=\int_AYdP \hspace{0.5cm} \forall A\in \sigma'$$ Maybe I'm wrong, but intuitively in $\sigma'$ all the events we 'can see' have probability 1, so conditioning by $\sigma'$ is like conditioning by the total set.

Answer 1

Integrating an integrable function on a set of zero measure gives $0$, hence the identity is satisfied whenever $A$ has measure $0$.

The other case is when $A$ has measure $1$. In that case, the LHS is the expectation of $Y$, and so is the RHS (because $\int_{\Omega\setminus N}Y\mathrm dP=\mathbb E[Y]$ if $\mathbb P(N)=0$.