In his "Relativity", Einstein explains that the ultimate Newtonian picture of the universe (matter in Euclidean space) would be one in which all mass were concentrated in a single area in an otherwise vast ocean of empty space- a universe which would grow increasingly "impoverished". He cites this as unsatisfactory.

Einstein later references the question of the mass density of the universe as a whole, indicating that if space is infinite, that is, if space is Euclidean or Quasi-Euclidean, then the density of the universe would be 0. However, he indicates that if we admit even a small average mass density for the universe as a whole then we necessarily admit that space is finite.

I don't understand why he says that any density value greater than zero for the universe would imply finite space. Yes it is true that if matter is finite and space is infinite then the density is 0, and that if space is finite and matter is finite then the density is greater than 0, but what if we were to admit infinite space and infinite matter? Would we not then admit a average density greater than 0 without resorting to the concept of finite space?

Later in "Geometry and experience" Einstein seems to indicate that there is another reason to believe that space is finite based on reasons other than his arguments using the concept of average density. There he says that "the latest results of relativity" indicate that the universe is probably finite and spherical. I am confused by this because Einstein seemed to say in "Relativity" that the General Theory of Relativity could imply am infinite "Quasi-Euclidean" space of the universe or a finite universe- thus he resorts to the density argument. What is Einstein citing here with respect to "the latest results"? Are there consequences of the theory of relativity which directly imply a finite universe? Is positive curvature of the universe implied by the EFEs for example?

As a note: the lecture "geometry and experience" was given by Einstein in 1921, while "Relativity" was published in the 40's I believe. Perhaps he thought some results had been arrived at only to latter find that they were unsatisfactory before publishing "Relativity".