Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly increasing in $x$.
Does this imply $I$ and $X$ are positively correlated?
Does this imply $F(x \ | \ I=1)$ first-order stochastic dominates $F(x \ | \ I=0)$? Could you provide a basic proof if it does?
If yes, does the implication still hold if $\text{Prob}(I=1 \ | \ X=x)$ is non-decreasing? If no, do additional conditions on $\text{Prob}(I=1 \ | \ X=x)$ such as continuity (or measurability etc.) help for the implication to hold?
Does the reverse implication hold?
My measure theory knowledge is weak so I'd also appreciate if you help putting down this question in a measure-theoretically well-defined sense.
Thank you.