For integrable X and a sub-sigma-algebra G, every version of E[X|G] must be G-measurable.

Four outcomes a,b,c,d are equally likely. Let A={a,b,c}, B={b,c} and G be generated by {A,A^c}. What is E[1_B|G]?

• 1/2 on A; 1/2 on A^c
• 1 on A; 0 on A^c
• 2/3 on A; 0 on A^c
• 0 on A; 2/3 on A^c

Four outcomes a,b,c,d are equally likely, with X-values 0,2,4,6. G has blocks {a,b} and {c,d}. The constant Y=3 has E[Y]=E[X], so it must be E[X|G].

Roll a fair six sided die. Given that the roll is even, what is the conditional expectation of the roll?

• 3
• 3.5
• 4

X takes -3,0,3 with probabilities 1/4,1/2,1/4. G records whether X=0. Since E[X|G]=0=E[X], X is independent of G.

For real X with E[X^2]<infinity, onto which space does E[X|G] project X in L2(F)?

• The closed subspace of G-measurable L2 variables.
• The collection of all G-events, viewed as subsets of Omega.
• The subspace of constant variables, for every choice of G.
• The space of all integrable F-measurable variables in L1.

Four outcomes are equally likely. X has values 1,3,5,7. G groups the first two and last two outcomes. Z is 2 on the first block and -1 on the second. Compute E[(X-E[X|G])Z].

If integrable X is independent of the sub-sigma-algebra G, then E[X|G]=E[X] almost surely.

Two proposed versions of the same E[X|G], with integrable X, disagree on a set of probability 1/2. What follows?

• Both satisfy the definition because versions may differ on positive-probability events.
• At least one does not satisfy the conditional expectation definition.
• G must be trivial because disagreement determines the information available.

For every integrable X and any sub-sigma-algebras G,H, E[E[X|G]|H]=E[X|H] almost surely, even when H is not contained in G.