[Math] Intution behind conditional expectation when sigma algebra isn’t generated by a partition

Question

I'm struggling with the concept of conditional expectation, when the sigma algebra on which it is conditioned isn't generated by a partition.

If $(\Omega,\mathcal{F},P)$ is a probability field such that $\mathcal{F}$ is generated by a partition $\Lambda_n$.

Then we know that:

$E[X|\mathcal{F}]$ = $E_1$$[X|\mathcal{F}]$ $I(\omega \in \Lambda_1)$ +$E_2$$[X|\mathcal{F}]$ $I(\omega \in \Lambda_2)$ + $E_3$$[X|\mathcal{F}]$ $I(\omega \in \Lambda_3)$ + …..

Where $E_i[.]$ is the expectation calculated as per the conditional probability $P(.|\Lambda_i)$

Hence when $\omega$ is in $\Lambda_i$ the conditional expectation gives the expectation of random variable X given that the observed event is $\Lambda_i$ and hence use the modified conditional probability rather than the original one. However this interpretation is only valid as long as the conditioning sigma algebra is generated by a partition. Is there a similar interpretation for a general case?

i.e what will it physically represent?

Any help will be greatly appreciated!

Thanks!

Best,
Adwait

Answer 1

The best intuition that I have for conditional expectation is that it's a projection. Also, try thinking about the conditional expectation as a Radon Nikodym derivative.