There are beings out there in other parts of Reality, who either anticipate seeing results with non-Born probabilities, or care about future alternatives in non-Born proportions. But (as I speculated earlier) those beings have much less measure under a complexity-based measure than us.

Or is there a sense in which we end up seeing results with a certain probability, a certain measure of ourselves going into those worlds, regardless of what we care about?

In other words, what you're asking is, is there is an objective measure over Reality, or is it just a matter of how much we care about about each part of it. I've switched positions on this several times, and I'm still undecided now. But here are my current thoughts.

First, considerations from algorithmic complexity suggest that the measure we use can't be completely arbitrary. For example, we certainly can't use one that takes an infinite amount of information to describe, since that wouldn't fit into our brain.

Next, it doesn't seem to make sense to assign zero measure to any part of Reality. Why should there be a part of it that we don't care about at all?

So that seems to narrow down the possibilities quite a bit, even if there is no objective measure. Maybe we can find other considerations to further narrow down the list of possibilities?

If you look at it closely, this is really about an instantaneous measure of the weight of experience, not about continuity between experiences.

I'd say that "continuity between experiences" is a separate problem. Even if the measure problem is solved, I might still be afraid to step into a transporter based on destructive scanning and reconstruction, and need to figure out whether I should edit that fear away, tell the FAI to avoid transporting me that way, or do something else.

But why don't the same arguments on continuity work on measure in general?

I don't understand this one. What "arguments on continuity" are you referring to?