The conditional probability of a continuous random variable $X$ given $Y=y$ is defined as
$1) \ E[X|Y=y] = \displaystyle \int x \frac{f_{X,Y}(x,y)}{f_{Y}(y)}dx $
My question is the following:
Does a definition similar to $(1)$ exist for $E[X|\mathcal{H}]$?
Where $X: \Omega \to \mathbb{R}^{n}$ is a random variable such that $E[|X|] < \infty$, and $\mathcal{H} \subset \mathcal{F}$ is a $\sigma$-algebra
I'm aware from wiki and other sources that by definition $E[X|\mathcal{H}]$ is the (a.s. unique) function from $\Omega$ to $ \mathbb{R}^{n}$ satisfying the following:
$a)$ $E[X|\mathcal{H}]$ is $\mathcal{H}$ - measurable
$b) $ $\int_{H}E[X|\mathcal{H}]dP = \int_{H}X dP$ for all $H \in \mathcal{H}$.
However I am still struggling to construct an intuitive understanding of what $E[X|\mathcal{H}]$ means in comparison to $(1)$.