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Recall that for events $A,B$ and $C$ with $P(C)>0$, $$ P(A \cap B \mid C) = P(A \mid B \cap C)\cdot P(B \mid C). $$

I'd like to show an analogous result when $P(C) = 0$ but am having trouble. The context is $C = \{X = x\}$ for a continuous random variable $X$. I'm familiar with regular conditional probabilities, but I can't seem to apply them correctly here (what is $P(A \mid \{X = x\}, B)$ in terms of regular conditional probabilities?)

To be clear, I'm trying to prove $$ P(A \cap B \mid \{X = x\}) = P(A \mid B \cap \{X = x\})\cdot P(B \mid \{X = x\}). $$

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    It might help to explain how you define P(A|B,X=x) when (X=x) has probability zero. – Did Sep 16 '15 at 06:33
  • To focus on the last line, your whole "probability world" has been restricted to the cases $X=x$. Do you have probability distributions for $A,B$ in this restricted circumstance? – hardmath Sep 16 '15 at 12:28
  • Follow @Did's lead. Take limits. – BruceET Sep 16 '15 at 16:05

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