I think that space is infinite. Space is just the void that our planet and every detectable and as of yet undetectable 'thing' resides in. It's like an ant climbing around on a leaf that's drifting in the current with a large tree in an infinite ocean. The ant is the human species, the leaf is earth, the tree is the universe and the ocean is space.
I think that the matter inside of space is finite--to an extent. There may be an infitine number of other trees floating around out there.
I think that just being able to somehow detect the edge of our own universe will be a big accomplishment. So far the trend is that the further back we look the newer everything is. Soon there's only going to be dust, which is hard to detect without an energy source being emitted from behind it.
If not for the space dust and innumerable other objects and debris and phenomenon, our sky would already shine like daytime. I read this somewhere, a search on star formation or something should give some info.