[cgmorton]

If the universe is infinite in size, does that automatically imply that it is infinite in mass?

That is, I can imagine a mass distribution that decreases as it moves away from the center, such that the total amount of mass is finite. In which case these problems don't apply.

I'm not sure whether that's possible though. If space-time is flat, that means there's a specific mass density in that universe, in which case infinite mass exists. For an infinite universe to have finite mass would mean a density that tends to zero, in which case we should observe a saddle shape in the experiments described. That is, unless space is just -locally- (in 14.6 billion light years) flat.

I don't claim to be right. But, you can use this possibility to ease your existential dread at being one of infinitely many near-identical copies. Of course you still have to deal with many-world quantum physics...