Probability Theory: Conditional Independence and Independence I have the following definition of conditional independence: $X$ and $Y$ are called conditionally independent given a sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$ if for all bounded Borel functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$, it holds: $E[f(X)g(Y)|\mathcal{G}]=E[f(X)|\mathcal{G}]E[g(Y)|\mathcal{G}]$.
However, I can not really get that.
In particular, what is really the difference between conditional independence and "unconditional independence"? Can someone maybe provide an example showing how and why these two things differ. Maybe an example that conditional independence is not implied by "unconditional/regular?!" independence? 
Thank you.

Consider $(X,Y)$ uniform on $\{-1,1\}^2$, then $(X,Y)$ is independent but is not independent conditionally on $Z=XY$. For example, $E(XY\mid Z)=Z$ while $E(X\mid Z)=E(Y\mid Z)=0$ hence $E(XY\mid Z)\ne E(X\mid Z)E(Y\mid Z)$.
Similar counterexamples show that conditional independence does not imply independence.
