Suppose G₂ ⊆ G₁ (G₂ is coarser than G₁). Which expression does the tower property say equals E[X|G₂]?
AE[E[X|G₂]|G₁]
BE[E[X|G₁]|G₂]
CE[X|G₁]
DE[X]
The tower property states E[E[X|G₁]|G₂] = E[X|G₂] when G₂ ⊆ G₁. Conditioning first on the finer algebra G₁ and then on the coarser G₂ is equivalent to conditioning on G₂ directly — the coarser conditioning 'wins.' Option A also equals E[X|G₂] because E[X|G₂] is already G₁-measurable, but option B is the direct statement of the tower property.
Question 2 True / False
E[X|G] is a fixed real number that represents the average of X given the information in G.
TTrue
FFalse
Answer: False
E[X|G] is a random variable — a measurable function on the sample space — not a number. It is G-measurable, meaning its value can vary depending on which outcome ω occurs. Only when you condition on a specific event (e.g., E[X|Y = y]) or take the expectation E[E[X|G]] = E[X] do you get a number.
Question 3 Short Answer
What two conditions must a random variable Z satisfy to be the conditional expectation E[X|G]?
Think about your answer, then reveal below.
Model answer: Z must be G-measurable, and it must satisfy E[Z · 1_A] = E[X · 1_A] for every A ∈ G.
G-measurability ensures Z only 'uses' information available in G. The integral condition E[Z · 1_A] = E[X · 1_A] encodes that Z correctly captures the average of X on every G-observable event. Together these two conditions uniquely determine E[X|G] (up to almost-sure equality), which is the content of the Radon-Nikodym theorem.