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

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 $$E[X_t + X_{t-1} | F_{t-1}]$$?

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.
$$E[X_{t-1} | F_{t-2}] + E[X_t | F_{t-1}] \neq E[X_t + X_{t-1} | F_{t-1}]$$
E.g. If $$X_t$$ is Brownian motion, the LHS equals $$X_{t-2} + X_{t-1}$$, whereas the RHS is $$2X_{t-1}$$.
• Brownian motion is a martingale, so $E(B_t|F_s) = B_s$ whenever $t>s$. @AlirezaBakhtiari Commented Oct 18, 2023 at 5:14