How to prove conditional independence property holds if A and B are just independent

Question

How do I prove given independent variables $A$ and $B$, that given $P(A \cap B) \mid F)$ is equal to $P(A \mid F) \cdot P(B\mid F)$

Answer 1

Suppose $\Pr(A) = \Pr(B) = 1/2$ and $A$ and $B$ are independent. Let the event $F$ be $\big[A \text{ or } B\big].$ Then $\Pr(A\mid F) = \Pr(B\mid F) = 2/3$ and $\Pr(A\cap B\mid F) = 1/3.$ And $$ \frac 2 3 \times \frac 2 3 = \frac 4 9 \ne \frac 3 9 = \frac 1 3. $$ So $A$ and $B$ are not conditionally independent given $F.$ $$ \begin{array}{r|cc} & B & \text{not }B \\ \hline A & 1/4 & 1/4 \\ \text{not } A & 1/4 & 1/4 \end{array} $$