Notation for Conditional Expectation I am kind of confused by the notations of conditional expectation.
Let $(\Omega, \mathcal F,\mathbb P)$ be the given probability space, $X$ be a random variable and $A \in \mathcal F$. 
I am confused with the notation $E[X|A]=\frac{\int_AXd\mathbb P}{\mathbb P(A)}$ since $E[X|A] \neq E[X|\sigma A]$.
But if so, why the notation like $E[X|A]$ is widely used? It is so confusing.
Can anyone explain the reason for using this notation? Thanks in advance.
 A: Unfortunately, standard notations for conditional expectations are a bit confusing. What is true is $E[X|\sigma (A)]$ is the random  variable which has the value $E[X|A]$ on $A$ and $E[X|A^{c}]$ on $A^{c}$. The concept of conditional expectation given a sigma algebra came later and things like $E[X|A]$ existed earlier and the mess has somehow been created. Let us live with it :-)
A: $\mathbb E[X\mid A]$ can be interpreted as the answer to the question: "what is the expectation of $X$ under the extra condition that event $A$ occurs?"
You write $\sigma A$ but formally that should be $\sigma(\{A\})$ which is the smallest $\sigma$-algebra on $\Omega$ that contains $\{A\}$ as subcollection (or equivalently contains $A$ as element). 
It is evident that: $$\sigma(\{A\})=\{\varnothing,A,A^{\complement},\Omega\}$$
Further $\mathbb E[X\mid\sigma(\{A\})]$ is formally a random variable that is measurable wrt $\sigma(\{A\})$ and satisfies the condition:$$\int_BX(\omega)\mathbb P(d\omega)=\int_BE[X\mid\sigma(\{A\})](\omega)\mathbb P(d\omega)\text{ for every }B\in\sigma(\{A\})\tag1$$
The fact that $\mathbb E[X\mid\sigma(\{A\})]$ is measurable wrt $\sigma(\{A\})$ reveals that we must have constants $c, d$ with:$$\mathbb E[X\mid\sigma(\{A\})]=c\mathbf1_A+d\mathbf1_{A^{\complement}}$$
Then substituting in $(1)$ for $B$ the $4$ elements of $\sigma(\{A\})$ we find the conditions:


*

*$0=0$

*$\int_AX(\omega)\mathbb P(d\omega)=c\mathbb P(A)$

*$\int_{A^{\complement}}X(\omega)\mathbb P(d\omega)=d\mathbb P(A^{\complement})$

*$\mathbb EX=c\mathbb P(A)+d\mathbb P(A^{\complement})$
Based on that we find $c=\mathbb E[X\mid A]$ as defined in your question and $d=\mathbb E[X\mid A^{\complement}]$ so that:$$\mathbb E[X\mid\sigma(\{A\})]=\mathbb E[X\mid A]\mathbf1_A+\mathbb E[X\mid A^{\complement}]\mathbf1_{A^{\complement}}$$
The last bullet also arises if we take expectation on both sides:$$\mathbb EX=\mathbb E[X\mid A]\mathbb P(A)+\mathbb E[X\mid A^{\complement}]\mathbb P(A^{\complement})$$
I hope this makes things more clear for you and also provides you a link between conditional expectation $\mathbb E[X\mid A]$ (a real number) and $\mathbb E[X\mid\sigma(\{A\})]$ (a random variable, which is actually the Radon-Nikodym derivative of $X$ wrt $\sigma$-algebra $\sigma(\{A\})$).
