Conditional expectation given an event is equivalent to conditional expectation given the sigma algebra generated by the event This problem is motivated by my self study of Cinlar's "Probability and Stochastics", it is Exercise 1.26 in chapter 4 (on conditioning).
The exercise goes as follows: Let H be an event and let $\mathcal{F} = \sigma H = \{\emptyset, H, H^c, \Omega\}.$ Show that $\mathbb{E}_\mathcal{F}(X) = \mathbb{E}_HX$ for all $\omega \in H.$
I'm not quite clear what I'm supposed to show, since when $\omega \in H$, then the $\sigma$-algebra is "reduced" to the event H, or am I misunderstanding something here?

That is really strange. How about showing the context? Anyway:
If $\mathbb{E}_\mathcal{F}(X) = \mathbb{E}(X | \mathcal{F})$
and if $\mathbb{E}_HX = \mathbb{E} [X|H]$,
then $\mathbb{E}(X | \mathcal{F}) = \mathbb{E}(X | H)1_H + \mathbb{E}(X | H^C)1_H^C$
If $\omega \in H$, then
$$\mathbb{E}(X | \mathcal{F})(\omega) = \mathbb{E}(X | H)1_H(\omega) + \mathbb{E}(X | H^C)1_H^C(\omega)$$
$$= \mathbb{E}(X | H)(1) + \mathbb{E}(X | H^C)(0)$$
$$= \mathbb{E}(X | H)$$
But you probably already knew that.
