Conditional Expectation of X

Question

How do I calculate the conditional expectation of

$E(X \mid X)$

where $E \vert X \vert<\infty$?

Answer 1

$\mathbb E\left(X\mid Y\right)$ is by definition a random variable satisfying:

• It is measurable with respect to the $\sigma$-algebra generated by rv $Y$.

• $\int_{A}X\left(\omega\right)\rm d P\left(\omega\right)=\int_{A}\mathbb E\left(X\mid Y\right)\left(\omega\right)\rm d P\left(\omega\right)$ for each set $A$ in $\sigma$-algebra generated by rv $Y$.

The first condition comes to the same as demanding that it can be written as $f\left(Y\right)$ for a Borel-measurable function $f:\mathbb{R}\rightarrow\mathbb{R}$.

In special case $\mathbb E\left(X\mid X\right)$ function $f=1_{\mathbb{R}}$ can do the job and $\mathbb E\left(X\mid X\right)=X$.