Notation and a problem with Aleatory variables (Advanced Probability) I am studying advanced probability and I have a question with notation. One exercise says: Let $(\Omega,B)$, show that $A \in B$ iff $1_A \in B$. But, $1_A$ is a function, what the book means with $1_A \in B$? Means that $1_A$ is a mensurable function? Another question is: Suppose $F(x) = P[X \le x]$ is continuos at $x$. Then $F(x)$ is mensurable and $Y = F(X)$ has uniform distribution, i.e. $$P[Y \le y] = y, ~ y \in [0,1].$$

As zoli answered above $1_A \in \mathcal{B}$ means that $1_A$ is a $\mathcal{B}$- measurable function.

I would like to address your second question:

If $F(x)$ is continuous, then it is clear that $F$ is measurable, but let's look at Y = F(X): for $a > 0$, the set $\\{x: F(x)
This is essentially the same idea of @zoli. I only wanted to avoid using $F^{-1}$ since the function might not be invertible.

**remark:** take $a_n \downarrow a$ to obtain $P(Y\leq a) = \lim_nP(Y \leq a_n) = \lim_n a_n = a$ this can also be used to the case $a = 0$