I have a conceptual problem with the relation between two integrable functions that are equal a.e.. Here there is a possible setting, to make things more concrete.
Question:
Assumptions:
- $(X, \Sigma, \mu)$ is a measure space,
- $A, B \in \Sigma$, with $A \subseteq B$,
- $\phi \in [0,1]^X$ is a measurable (and integrable) function, with $\phi (X \setminus A) = \{ 0 \}$.
If we focus on the conditional expectation of $\mu$, defined for an arbitrary $Y \in \Sigma$ as $$ \mathbb{E} \ (\ \phi \ | \ Y ) := \frac{1}{\mu (Y)} \int_Y \phi d \mu, $$ can we state – given the previously described setting – that the following is true: $$ \frac{\mathbb{E} \ (\ \phi \ | \ B ) }{\mathbb{E} \ (\ \phi \ | \ A )}= 1 ?$$
To me this make sense, because essentially the two expectations are the same, given the condition $\phi (X \setminus A)$. Still I am not completely sure.
Is the line of reasoning sound?
Is actually correct to talk about equality a.e. in this kind of context?
Thank you for your time.
Your conditions do not happen in useful situations. Apply your hypothesis to $A=\emptyset$ and $B=X.$ The $\phi$ must be zero on the complement of $A,$ i.e., everywhere.
Now if these conditions apply only for a particular choice of $A$ and $B$ then I have to argue that $E[\phi|A]$ is different from $E[\phi|B]$ as soon as one of these two numbers is nonzero and $\mu(B-A)>0$ (so that $\mu(A)\neq\mu(B)$).