$\sigma$-field generated by random variable

Question

If we have $$ P(Y_i=1)=\frac{1}{2} , \, P(Y_i=-1)=\frac{1}{2} \text{ for } i=1,2, $$ and describe $$X=Y_1+Y_2$$ then what is $\sigma$-field $X$ ? Thanks a lot.

Answer 1

If $X$ denotes a random variable defined on probability space $\langle\Omega,\mathcal A\rangle$ then: $$\sigma(X)=\{\{X\in B\}\mid B\in\mathcal B\}$$ where $\mathcal B$ denotes the $\sigma$-algebra of Borel sets on $\mathbb R$.

If moreover $\{X\in C\}=\Omega$ where $C$ denotes a countable subset of $\mathbb R$ then: $$\sigma(X)=\{\{X\in D\}\mid D\subseteq C\}\tag1$$

This because $\{X\in D\}\in\sigma(X)$ for every $D\subseteq C$, and conversely the RHS of $(1)$ is evidently a $\sigma$-algebra that contains every set $\{X\in B\}=\{X\in B\cap C\}$ for $B\in\mathcal B$.

Further the countability of $D$ tells us that: $$\{X\in D\}=\bigcup_{d\in D}\{X=d\}$$

In your case there is actually not enough information.

Things would be different under the extra conditions that $\{Y_1\in\{-1,1\}\}=\Omega=\{Y_2\in\{-1,1\}\}$.

So not $Y_i\in\{-1,1\}$ almost surely, but instead $Y_i(\omega)\in\{-1,1\}$ for every $\omega\in\Omega$.

Under that condition we have $\{X\in C\}=\Omega$ for $C=\{-2,0,2\}$.

Note that this set has $8$ subsets $D$ and for every such $D$ we can find an expression for $\{X\in D\}$ in terms of the $Y_i$.

For instance $\{X\in\{-2,2\}\}=\{Y_1=Y_2\}$.