I should point out here that the space you're averaging over isn't the space of actions you can take, it's the space of states-of-the-world.

Now arguably that could be taken to be finite too, and that avoids these problems. Still, I'm quite wary. The use of surreals in particular seems pretty unmotivated here. It's tossing in the kitchen sink and introducing a whole host of potential problems just to get a few nice properties.

(I would insist that utilities should in fact be bounded, but that's a separate argument...)