conditioning on an independent sigma-algebra

Some information is simply useless for predicting X. If what G tells you has nothing to do with X — they are statistically independent — then learning G should not move your estimate of X at all. The rule makes this exact: conditioning on independent information collapses the conditional expectation back to the plain, unconditional one. Knowing the result of a coin flip in Tokyo does not change your best guess of tomorrow's weather in Lisbon.

Precisely: if X is independent of the sigma-algebra G (meaning X is independent of every event in G), then E[X given G] = E[X], a constant. You can verify it against the two defining properties: a constant is trivially G-measurable, and by independence the integral of X over any A in G is E[X] times P(A) = integral of the constant E[X] over A, so partial averaging holds. Both demands are met, so by uniqueness the conditional expectation must be that constant. Conditioning on something irrelevant gives you nothing new — exactly as intuition insists.

This is one of the two pillars of conditional reasoning, the perfect complement to 'taking out what is known'. Together they handle the typical martingale step E[ X_{n+1} given F_n ]: the past F_n that already determines part of X_{n+1} is pulled out as known, while the fresh increment, independent of F_n, has its conditioning dropped and reverts to its plain mean. One honest caution: the condition is genuine independence, not mere zero correlation. A G that is uncorrelated with X but not independent can still shift E[X given G] away from E[X] — zero correlation only kills LINEAR dependence, and conditional expectation sees nonlinear dependence too.