Understanding meaning of condition in $E(X|Y,Z)$ I have a problem understanding the meaning of condition in expression: $E(X|Y,Z)$, where $X,Y,Z$ are random variables. 
I only find some formulas of consistency and that $E(X|Y,Z)$ is equivalent to $E(X|(Y,Z))$. 
Explanations about $\sigma$ algebras are not very intuitive for me. Help!
 A: Hoi boy, 
First it s a very hard notion to get familiar with and it will take you even years to deeply understand it.
Second I would say that if  you don't understand/ are familiar with $\sigma$ algebra then it makes it hard to deeply understand the conditional expectation as it's directly linked to that.
You can understand it as an orthogonal projection of $X$ (if you are working with variables in $\mathbb{L^2(\Omega,\mathcal{F},\mathbb{P})}$) onto the minimal space that makes that Y and Z are measurable. In that case we can state that : The conditional expectation is unique, exists and is exactly the orthogonal projection (for the scalar produciot $<Z_1,Z_2> = \mathbb{E} [Z_1Z_2]$).
Indeed for instance any measurable function of Y and Z would stay the same meaning : $\mathbb{E}[f(Y,Z) | Y,Z ] = f(Y,Z)$ no projection since it's already in the space $\sigma(Y,Z)$.
This formula is also intuitive : if $\mathcal{H}$ included in $\mathcal{G}$ then projecting first on $\mathcal{G}$ then on $\mathcal{H}$ is the same as projecting on $\mathcal{G}$ as stand the following formula : $\mathbb{E}[\mathbb{E}[X|\mathcal{G}]|\mathcal{H}] = \mathbb{E}[X|\mathcal{H}]$ so for instance it can give the following for your example :  $\mathbb{E}[\mathbb{E}[X|Z]|Y,Z] = \mathbb{E}[X|Y,Z]$
