Union of sigma fields generated by random variables

Question

Let $X,Y:(\Omega,\mathcal{F},\mathbb{P}) \rightarrow (\mathbb{R},\mathbb{B}_{\mathbb{R}})$ , does that necessarily mean $\sigma(X) \cup \sigma(Y)$ is a sigma field? Generally,it is not true that the union of two sigma fields is a sigma field.But what about this case?

Answer 1

Take $(\Omega,\mathcal F) = (\mathbb R, \mathcal B( \mathbb R))$ with some probability measure $\mathbb P$

Take $X,Y$ as $\chi_A, \chi_B$ respectivelly for some borel sets $A,B$, where $\chi_A(x) = 1$ if $x \in A$ and $0$ otherwise.

Then $\sigma(X) = \{ X^{-1}(C) : C \in \mathcal B(\mathbb R) \}$, but since $X$ can attain only values $0$ and $1$ we have:

$X^{-1}(C) = A$ when $1 \in C, 0 \not \in C$

$X^{-1}(C) = A'$ when $1 \not \in C, 0 \in C$

$X^{-1}(C) = \emptyset$ when $1,0 \not \in C$

$X^{-1}(C) = \mathbb R$ when $1,0 \in C$

So $\sigma(X) = \{ \emptyset, \mathbb R, A , A'\}$ and similarly $\sigma(Y) = \{ \emptyset, \mathbb R, B, B'\}$

When for example $A = (0,1)$, $B = (1,2)$, then $A,B \in \sigma(X) \cup \sigma(Y)$ but $A \cup B \not \in \sigma(X) \cup \sigma(Y)$