Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.


I think it should be as follows: 


Let $(X,\mathcal{B},\nu)$ be a measure space and let $\mathcal{F}\subset\mathcal{B}$ a sub$-\sigma-$algebra, such that $\nu$ is $\sigma-$finite in $\mathcal{F}$. Then for all $f\in L^1(X,\mathcal{B},\nu)$ there exists $g\in L^1(X,\mathcal{F},\nu|\_{\mathcal{F}})$ such that
$$\int\_{E}fd\nu=\int\_Egd\nu|\_{\mathcal{F}},\qquad\forall E\in\mathcal{F};$$
then $g=:\mathbb{E}\_{\nu}[f|\mathcal{F}]$ is called the conditional expectation of $f$ given $\mathcal{F}$.


Is this the correct way to define conditional expectation? There is another way to define it without requiring the hypothesis that $\nu$ be $\sigma$-finite in $\mathcal{F}$?



Conditional Expectation for $\sigma$-finite measures
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The definition you quoted is correct. 


However, there can be no reasonable notion of the conditional expectation without the sigma-finiteness condition, even in the discrete setting. E.g., let $X=\N$, $\B=2^\N$, and let $\F$ be any sigma-algebra over $\N$ containing an infinite atom $A\subseteq\N$; for instance, one may take $\F=\{\emptyset,\N\}$, with $A=\N$. Let $\nu$ be the counting measure on $\B=2^\N$, and let $f(x)=1/2^x$ for $x\in\N$. Then $E\_\nu f=1\in\R$.


However, on the atom $A$ one cannot reasonably ascribe any value to the conditional expectation $E\_\nu(f|\F)$, because such a value (say $v$) could reasonably be only the $\nu$-average of $f$ on $A$. Indeed, if you take $v=0$, this would imply $\int\_A f\,d\nu=0$, which is false; if you take $v\ne0$, this would imply $|\int\_A f\,d\nu|=|v|\nu(A)=\infty$, which is also false. 


The problem here is that, while the measure $\nu$ is sigma-finite, its restriction $\nu|\_\F$ to $\F$ is not.


