How is summation defined for two measurable functions with different measurable sets?

Question

I have a question about the definition of sums of RVs and I can't find the answer to it.

Assume that $A$ and $B$ are $\sigma$-algebras and $A \subset B$, if $X$ is $A$-measurable and $Y$ is $B$-measurable then how is $X+Y$ is defined? My guess is that it's defined as $X + Y$ as a $A$-measurable function, but I'm not sure if that's the case or even whether I'm asking the correct question.

In case you wonder about the context, it comes up when I wanted to reason about the sums of conditional expectations in a filtration. For instance assume $X_t$ is adapted to $F_t$, then how would you define $E[X_{t-1}|F_{t-2}]$ + $E[X_t|F_{t-1}]$? Is it $E[X_t + X_{t-1} | F_{t-1}]$?

Answer 1

To make sense of the measurability of $X+Y$, then you must be operating under the same underlying probability space. When dealing with a filtration $\mathcal{F}_t$, you will always have an underlying $\sigma$-algebra $\mathcal{F}$ for which $\mathcal{F}_t$ is a sub-$\sigma$-algebra for each $t$, so you will be able to make sense of the measurability of $X+Y$ relative to a $\sigma$-algebra of your choice.

For your question about conditional expectations, in general

$$E[X_{t-1} | F_{t-2}] + E[X_t | F_{t-1}] \neq E[X_t + X_{t-1} | F_{t-1}]$$

E.g. If $X_t$ is Brownian motion, the LHS equals $X_{t-2} + X_{t-1}$, whereas the RHS is $2X_{t-1}$.