I visualize everything is contained within an immeasurable ice sphere. Whenever new 'space' is needed to contain new matter, the inner surface of the ice sphere melts some more. Ice has more volume than water, so when the ice turns to water, there is a little left over space.
I was kind of surprised to see that picture, as it nearly matches my crude model.
Kind of a Steady State model there. As to empty space, if you can measure it that would mean you are putting something in it to measure with so it isn't empty anymore. The ice sphere might be an old style black hole so it wouldn't radiate anything and there could be truly empty space outside although by saying outside we are putting a determination on the empty space and for that very action it isn't empty anymore.