sigma-algebra generated by a r.v.

Hallo,

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

I.e. I want to show

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

Obviously, it holds that

$\displaystyle \sigma(f(X))\subseteq \sigma(X) \subseteq \sigma(X,f(X)).$

Can anybody help me?

Thanks in advance.

Re: sigma-algebra generated by a r.v.

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