Is sigma algebra $\sigma(X,Y,X+Y)=\sigma(X,Y)$?

Question

The question above challenged my knowledge of probability theory.

I knew that $\sigma(X + Y) \subset \sigma(X, Y)$, discussion of which can be found for example here

Is it true that $\sigma(X+Y) \subset \sigma(X,Y)$?

My intuition for $$ \sigma(X,Y,X+Y)=\sigma(X,Y) $$ is that $\sigma(X,Y)$ completely determines $(X+Y)$, thus $\sigma(X,Y,X+Y)$ can not have any additional information not contained in $\sigma(X,Y)$. Is this intuition correct?

If it is correct, than how one would formalize this?

Answer 1

For two random variables, can you say that $\sigma(X,Y) = \sigma(\sigma(X),\sigma(Y))$?

If so, this finishes the proof because as you said, $\sigma(X,Y) \subset \sigma(X,Y)$, and then $$\sigma(X,Y) \subset \sigma(X,Y,X+Y) = \sigma(\sigma(X,Y), \sigma(X+Y)) \subset \sigma(X,Y)$$