# Math Help - sigma-algebra generated by a r.v.

1. ## sigma-algebra generated by a r.v.

Hallo,

How can I show that the $\sigma-$algebra generated by a random variable $X$ and the $\sigma-$algebra generated by $X$ and $f(X)$ ( $f$ is an appropriate measurble function) are equal?

I.e. I want to show

$\sigma(X)=\sigma(X,f(X)).$

Obviously, it holds that
$\sigma(f(X))\subseteq \sigma(X) \subseteq \sigma(X,f(X)).$

Can anybody help me?
Show that $\sigma(X)$ contains all the elements of the form $X^{-1}(B)$ and $(f\circ X)^{-1}(B)$, where $B$ is measurable.