I am currently looking into conditional expectations (CE) and how to compute them. I know that in the typical finite-dimensional, continuous settings, one would do the computations via densities. However, I was wondering how, if possible, one would do that without them. So the question is
Let $(\Omega, \mathcal{A},\text{P})$ be a probability space and let $\mathcal{B} \subset \mathcal{A}$ be a sub-$\sigma$-algebra. Let $X: \Omega \rightarrow \mathbb{R}$ be a random variable, how do I compute the $\mathcal{B}$-measurable map
\begin{align*} \mathbb{E}(X | \mathcal{B}) : \Omega \rightarrow \mathbb{R} \end{align*}
without making use or knowing densities?
The only characterization and approach I have that comes to mind is the orthogonality relation: $\mathbb{E}(X | \mathcal{B})$ is an orthogonal projection from $L^2 (\mathcal{A})$ onto $L^2(\mathcal{B})$. This seems very tricky though for practical computations as one changes the measurability.