When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is $$ E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\mathscr D_j)I_{D_j} $$ where $I_{\mathscr D_j}$ is an indicator function.
When random variable $X$ is continuous, The conditional expectation of $X$ with respect to a $\sigma$-algebra $\mathscr F$ is a nonnegative random variable $\def\E{\mathbf E}\E(X\mid \mathscr F)$, such that
- $ \def\E{\mathbf E}\E(X\mid \mathscr F)$ is $\mathscr F-measurable$
- for all $A \in \mathscr F\\$, $$\int_A \def\E{\mathbf E}\E(X\mid \mathscr F)\, dP=\int_A X\, dP$$
I find it is hard to link those two definitions. In discrete case, conditional probability is defined first and we derive conditional expectation based on conditional probability. And it is a similar way as to define conditional expectation w.r.t random variable. But in continuous case, we first state that there exists a unique random variable, and just give it a name as $\def\E{\mathbf E}\E(X\mid \mathscr F)$. I wonder why this random variable can be viewed as a conditional expectation in continuous case and its link to discrete case.
