conditional expectation given a sigma-algebra

Conditioning on a single variable Y is fine, but in a stochastic process you want to condition on 'everything observable so far' — a whole bundle of information that no single variable captures. The clean way to package 'a body of information' is a sigma-algebra G: a collection of events whose answers (happened or not) you are allowed to know. Conditioning on G means computing your best estimate of X using exactly the knowledge in G, no more and no less.

The result, written E[X given G], is again a random variable, and it is pinned down by two demands. First, it must be G-measurable: it may only depend on the information in G, so it cannot 'peek' at anything outside G. Second, it must match X on average over every event in G: for each set A in G, the average of E[X given G] over A equals the average of X over A (the partial-averaging condition). Remarkably, these two requirements determine E[X given G] uniquely (up to events of probability zero). Existence is guaranteed for any integrable X — you can build it as the Radon-Nikodym derivative of the measure 'integral of X over A' with respect to P restricted to G, or, when X has finite variance, as the orthogonal projection of X onto the space of G-measurable functions.

This abstract definition looks austere but it unifies everything: E[X given Y] is just E[X given G] where G is the information generated by Y, and ordinary expectation E[X] is the case where G is the trivial sigma-algebra (you know nothing). It is the engine under martingales, filtrations, and the whole machinery of stochastic processes. The honest caveat: E[X given G] is defined only up to a set of probability zero, so it is really an equivalence class of random variables, and all the standard properties (tower, linearity, and the rest) hold 'almost surely', not for every single outcome.