Goodnight

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 like that,

Let $(X,\mathcal{B},\nu)$ a measure space and $\mathcal{F}\subset\mathcal{B}$ a sub$-\sigma-$algebra, such that $\nu$ is $\sigma-$finita in $\mathcal{F}$. Then for all $f\in L^1(X,\mathcal{B},\nu)$ exists $g\in L^1(X,\mathcal{F},\nu|_{\mathcal{F}})$ such that $$\int_{E}fd\nu=\int_Egd\nu|_{\mathcal{F}},\qquad\forall E\in\mathcal{F},$$

$g(x):=\mathbb{E}_{\nu}[f|\mathcal{F}](x)$ it is called conditional expectation.

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

Thank you

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 $\mathcal{F}$?