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Suppose that $X,Y$ are random variables or events. I am wondering when $P(X) \leq P(Y)$ implies $P(X|A) \leq P(Y|A)$ for an event or random variable $A$.

One way $P(X) \leq P(Y)$ occurs is if for the events $\{X\}$ and $\{Y\}$,

$$ \{X\} \subset \{Y\}. $$

Given $\{X\} \subset \{Y\}$, does this imply $\{X|A\} \subset \{Y|A\}$? This seems like a massive abuse of notation so would this be an indicator function?

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    There are some massive conceptual misunderstandings at work here: note that $\{X\}$ and $\{Y\}$ exist but are not events (and that $\{X\} \subset \{Y\}$ only occurs when $X=Y$...) and that $\{X|A\}$ and $\{Y|A\}$ simply do not exist. – 2017-02-27

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First, we know that $$P(X|A) = \frac{P(X,A)}{P(A)}$$ and $$P(Y|A) = \frac{P(Y,A)}{P(A)},$$ thus $$P(X|A) \leq P(Y|A) \iff P(X,A) \leq P(Y,A).$$

So, if $$\{\omega : \omega \in X\} \subset \{\omega : \omega \in Y\}$$ then we clearly have $$\{\omega : \omega \in X\cap A\} \subset \{\omega : \omega \in Y \cap A\} \Rightarrow P(X,A)\leq P(Y,A) \iff P(X|A) \leq P(Y|A)$$ as desired. So that is a suffient condition and not just an "abuse of notation."

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    Why write $$\{\omega : \omega \in X\} \subset \{\omega : \omega \in Y\}$$ instead of $$X\subset Y$$ and $$\{\omega : \omega \in X\cap A\} \subset \{\omega : \omega \in Y \cap A\}$$ instead of $$X\cap A\subset Y\cap A$$ although these are strictly equivalent? I am asking you this because the OP seems to be lost in such notations (see my comment on main) hence being crystal clear on these set theoretical basics would definitely be a plus to them... – 2017-02-27
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    @did fair point. I thought it would be clearer, but perhaps I was wrong. Let me rethink how I should word this... – 2017-02-27
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When $X\subset Y, P(A)\neq 0$, we can check that $P(X|A) = \dfrac{P(XA)}{P(A)}\leq\dfrac{P(YA)}{P(A)} = P(Y|A)$.