Let ( <mi mathvariant="normal">Σ , F , P ) be a probability

Question

Let

(Σ, F, P)

be a probability space, let

X

be an

F

-random variable, let

G

be a sub-sigma algebra of

F

, let

ω \in Ω

, let

A

be the intersection of all the sets in

G

that contain

ω

, and suppose that

A \in F

. Then is it true that

E (X | G) (ω) = E (X | A)

?
I just wrote this statement down based on my intuition of what conditional expectation means, and I want to verify if it’s actually true. Note that sigma algebras need not be closed under uncountable intersections, so

A

need not be an element of

G

. But I assumed

A

is at least an element of

F

so that

E (X | A)

is meaningful.

Answer 1

No, when you condition on a sigma field, there is little connection to the basic definition of conditionings. The only result I know is when you condition on a discrete random variable. In other words, let

Z

be an integrable random variable and let

X

be a discrete random variable. Suppose all the values

X

take with positive probability are

(x_{k}) .

Then

E (Z ∣ X) = \sum_{k} E (Z ∣ X = x_{k}) 1_{{X = x_{k}}}

and here

E (Z ∣ X = x_{k})

is the usual conditional expectation over an event

A

with positive probabiltiy

E (Z ∣ A) = \frac{1}{P (A)} \int_{A} Z d P .

(Note that when

Z = 1_{{Y \in B}}

then you have

E (Z ∣ A) = \frac{P (Y \in B, X = x_{k})}{P (X = x_{k})} = P (Y \in B ∣ X = x_{k}) .)

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