In E. Carnal "Markov Properties for Certain Random Fields", there is a lemma 2.2 (i) which states:
If $\mathscr{A}\perp\mathscr{B}\mid\mathscr{C}$, then $\mathscr{A}\vee\mathscr{C}\perp\mathscr{B}\vee\mathscr{C}\mid\mathscr{C}$, where $\mathscr{A}\vee\mathscr{C}=\sigma(\mathscr{A}\cup\mathscr{C})$.
I want to prove this. I notice:
(1) It suffices to prove $\mathscr{A}\vee\mathscr{C}\perp\mathscr{B}\mid\mathscr{C}$.
(2) For any $\Lambda\in\mathscr{K}:=\{A\cap C:A\in\mathscr{A},C\in\mathscr{C}\}\subseteq\mathscr{A}\vee\mathscr{C}$, it holds that $$\mathbb{P}\{\Lambda\cap B\mid\mathscr{C}\}=\mathbb{P}\{\Lambda\mid\mathscr{C}\}\mathbb{P}\{B\mid\mathscr{C}\},\quad\forall B\in\mathscr{B}.$$ (3) Let $\mathscr{H}:=\{H\in\mathscr{A}\vee\mathscr{C}: \mathbb{P}\{H\cap B\mid\mathscr{C}\}=\mathbb{P}\{H\mid\mathscr{C}\}\mathbb{P}\{B\mid\mathscr{C}\},\forall B\in\mathscr{B}\}\supseteq\mathscr{K}$, it suffices to show that $\mathscr{H}$ is a $\sigma$-algebra, in that $\mathscr{H}\supseteq\sigma(\mathscr{K})=\mathscr{A}\vee\mathscr{C}$ $\Rightarrow$ $\mathscr{H}=\mathscr{A}\vee\mathscr{C}$.
(4) To show that $\mathscr{H}$ is a $\sigma$-algebra, I tried to prove $\mathscr{H}$ is both a $\pi$-system and a $\lambda$-system, but I failed on the former one.
Any new insights? Many Thanks!!!
Sorry, I made a holy mistake.
Instead of proving $\mathscr{H}$ is a $\pi$-system, I shall prove that $\mathscr{K}$ is a $\pi$-system, which is manifest.
Then, by $\pi$-$\lambda$ theorem, the conclusion $\mathscr{H}\supseteq\sigma(\mathscr{K})$ holds immediately.
(I should not prove that $\mathscr{H}$ is a $\sigma$-field, and I was mislead by the form of the conclusion $\mathscr{H}\supseteq\sigma(\mathscr{K})$.)