Let $A,B,C$ be measurable sets on a probability space. I'm trying to understand the meaning of the sentence: A conditioned on B is independent of C.
The conditional probability was defined as: $$ P(A|B)\equiv\frac{P(A\cap B)}{P(B)} $$ when $P(B)>0$.
Given the definition above, how should I interpret that sentence? Does that mean that $P\left(\left(A|B\right)\cap C\right)=P(A|B)P(C)$? If so, what does $(A|B)\cap C$ means?
Thanks for helping! :D
"A is independent of C", $A\perp C$, means that: $\mathsf P(A\cap C) = \mathsf P(A)\;\mathsf P(C)$
Likewise, the meaning of "A conditioned on B is independent of C," is that "A is conditionally independent of C, when given B".
$$A\perp C \mid B \;\iff\; \mathsf P(A\cap C\mid B) = \mathsf P(A\mid B)\;\mathsf P(C\mid B)$$
Nothing. It's nonsense; not well formed at all. The condition must apply to everything. It's the condition the probabilities are being measured over.
"Whether I take my umbrella to work, conditioned on it being a work day, is independent of whether it rains." We're not interested in the probability of whether it rains on weekends (and clearly those are days we don't take umbrella to work, so how can it be independent?). We're only interested in the probabilities on the given condition.
$$U\perp R \mid W \;\iff\; \mathsf P(U\cap R\mid W) = \mathsf P(U\mid W)\;\mathsf P(R\mid W)$$