independent random variable I have a question about independence of two random variables. Say X and Y are two random variables, if I know that X is independent of I(Y>y), for any real value y, where I() is the indicator function. Can we conclude that X is independent of Y ? How to understand this problem based on measure theory? 
 A: One definition of $X$ and $Y$ being independent is that $P(X > a \land Y > b) = P(X > a)P(Y > b)$ for any $a, b$. So you may conclude independence using what you have:
$$P(X > a \land Y > b) = P(X > a \land I(Y > b) > \frac 1 2) = \\=P(X > a)P(I(Y > b) > \frac 1 2) = P(X > a)P(Y > b)$$
Regarding your latter question: $P(X > a)$ can be interpreted, for instance, as the (probability) measure of the subset $\{\omega \in \Omega : X(\omega) > a\}$ of the probability space $\Omega$. The other expressions work similarly. Does this answer your question?
A: QUOTE:

if I know that X is independent of I(Y>y), for any real value y, where I() is the indicator function. Can we conclude that X is independent of Y ?

END QUOTE
This is ambiguous:

If there is any value of $y$ for which the indicator variable $I(Y>y)$ is independent of $X$, then . . . .

versus

If it is the caswe that for any value of $y$, the indicator variable $I(Y>y)$ is independent of $X$, then . . . .

There is a difference in meaning.  Just changing "any" to "every" in your original phrasing fully disambiguates it.
And then the bottom line is "yes".
