The proof of $F_{X|A}(-\infty) = 0$ and $F_{X|A}(\infty) = 1$

Question

How to prove this cumulative distribution function ?

$F_{X|A}(-\infty) = 0$ and $F_{X|A}(\infty) = 1$ ,

What does $F_{X|A}(-\infty)$ mean?Is it the same as $F_{X|A}(X=-\infty|A=-\infty)$?I have no idea how to begin proving this.

Answer 1

My guess is $F_{X|A} (-\infty)$ stands for $\lim_{x \to -\infty} \frac {P\{X \leq x, A\}} {P(A)} $. The events $\{X \leq x, A\}$ decrease to the empty set as $x$ decreases to $-\infty$ so their probabilities $\to 0$. Similarly, $\{X \leq x, A\}$ increase to $A$ as $x$ increases to $\infty$ so the probability $\to P(A)$. Now divide by $P(A)$.