Independence of intersections Let $(\Omega,\mathfrak{A},P)$ be a probability space and $A,B,C\in\mathfrak{A}$ some events where $A$ and $B$ are independent. I'm a bit confused now as I intuitively think that this also implies independence of $A\cap C$ and $B\cap C$. Am I right? Unfortunately, I immediately get stuck when I try to use the definition.

Consider $A = \{ 1,2 \}$, $B = \{1,4\}$ and $C = \{1\}$ on $\Omega = \{1,2,3,4\}$, with $P( X ) = \frac{|X|}{|\Omega|}$
$$P(A \cap B ) = 1/4$$
$$P(A) P(B) = 1/4$$ 
So A and B are independent, yet 
$$P( (A \cap C) \cap (B\cap C) ) = 1/4$$
But, note that
$$P(A \cap C) P (B\cap C)  = 1/16.$$
Therefore, these events are not independent
