Consider the probability space $([0,1]; B[0,1], L)$, where $B[0,1]$ contains the Borel sets intersecting $[0,1]$ and $L$ is the Lebesgue measure. How do I find the sigma algebra generated by a random variable defined on this space, $X = 1_{[0,1/2]}$? Secondly, how do I determine whether random variables defined on this space are independent or not, e.g., are $X = 1_{[0,1/2]}$ and $Y = 1_{[1/4,3/4]}$ independent?
For finding sigma algebra generated by $X$, I find $X^{-1}(1_{[0,1/2]})$ but what will be this inverse in Borel sets? For finding independence, it should be sufficient to show the sigma algebras generated by $X$ and $Y$ are independent right?