Conditional Expectation Given a Sigma-Algebra
Define conditioning on information rather than on an event, as the best prediction measurable with respect to what is known.
What a learner can do afterwards
- State the defining property and prove uniqueness up to a null set
- Recover elementary conditional expectation as the case of a finite partition
- Use the tower property and the projection interpretation on a worked example
1 · Read
Conditioning on a sigma algebra means conditioning on information rather than on a single event. The conditional expectation E of X given G is a G measurable forecast: it uses only what G knows. Its defining property is that it matches X in the mean on every G set, through E of indicator A times E of X given G equals E of indicator A times X. Two versions can differ only on null sets, which integrals cannot tell apart, so uniqueness holds up to null sets.
Roll a fair six sided die and let G record only whether the roll is even or odd. On the even block the forecast is the local mean 2 plus 4 plus 6 over 3, which is 4. On the odd block it is 1 plus 3 plus 5 over 3, which is 3. That is the finite partition case in general: condition on each block and report the average within it.
The tower property says averaging the forecast recovers the plain mean: E of E of X given G equals E of X. With two fair coin flips where G records only the first flip, the forecast averages back to 1, the mean number of heads. When X is independent of G, conditioning changes nothing and E of X given G equals E of X. The projection view says the same thing geometrically: E of X given G is the closest G measurable guess in mean square.
When you verify a candidate forecast, check the two halves separately: measurability with respect to G, then the defining identity on every G set. Students usually check one half and forget the other. If either half fails, the candidate is not the conditional expectation.
Condition on what you know, match the mean on every known set, and average the forecast to recover the mean.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.