I don't think that's established. We know the universe must have been far, far, more dense than it is now just at and just after the big bang. But I don't think we can go so far to say with certainty that it was finite, as measured from within its own construct.
The general consensus now is not that the universe expanded into some sort of preexisting space at the time of the big bang, but rather the 3 dimensions of space themselves came out of the big bang. My point is that the universe, even at and just after the big bang, is not like something you could measure from the outside, like measuring the diameter of an apple, being outside of the apple. Everything is inside -- even infinity itself. Not only is it not possible to look at it from the outside, it doesn't even make any sense.
It kind of reminds me of an old riddle. "If a plane crashes exactly on the boarder of the USA and Canada, where are the survivors legally buried?" The answer to the riddle is that the survivors are not buried. They didn't die (being survivors), so it doesn't even make any sense to discuss where they are buried. When discussing what was outside the universe at or before the big bang, many scientists reply 'nothing'. But it is not the kind of 'nothing' as in empty-space nothing. Instead it's like the 'undefined' kind of nothing. The kind of nothing that doesn't even make logical sense to talk about.
Even if one adopts the idea of the universe being a 0-dimensional point singularity at the moment of the big bang, that singularity wasn't sitting inside of some area of space. that singularity was still all of space itself and everything in it.
Maybe the universe (entire universe) is finite. I don't know. I just don't think we've established that it is. That's my only point.
Yes, that much we can say. The observable universe is and was finite.
