In accordance with the FLRW Metric with a curverature of $k=0$ (as observationally supported by several of NASA's experiments including WMAP, Planck satellite, DASI, etc.) the universe is spatially infinite. Of course, our observable universe is finite, but as far as we know, there is simply no edge to the universe. Does this imply that there is a non-finite amount of mass in the isotropic and homogeneous universe, assuming a Euclidean geometry?
The question is complicated by a lack of vocabulary for many features of the universe. The universe is thought to be "boundless" but finite. But that is not the same thing as "infinite". It simply has no edges or ends. That is a consequence more of the effect of curved space than of anything else. If you try to travel to the "edge" of the universe, you eventually return to where you started. That is because it is impossible for straight lines to exist in curved space and also impossible to even determine if you are travelling a straight line. All measures of your trajectory are also bent. That is why the world is better described by Riemannian or Minkowsky space. In both, the Euclidean idea that parallel lines never meet is no longer true. Neither supports the existence of parallel lines.
Like singularities in relativity, infinities most likely do not exist in any form. Relativity is a classical theory and has the limitations of one. Where "singularities" exist in General Relativity, the theory and math break down. Divisions by 0 do not give infinities except in peoples' wishes. In any case, the use of imaginary time rather than real time (Wick Rotation) reduces a Minkowsky space problem in D dimensions plus time to one in Euclidean space of D+1 dimensions. The singularities then do not exist, nor do the infinities. Calculations to the moment of the Big Bang and before then appear possible. It may even be possible to determine why the universe began if details of the particle physics can be worked out.
The volume of the universe is infinite, therefore there must be an infinite number of worlds. But not all of them are populated; therefore only a finite number are. Any finite number divided by infinity is as close to zero as makes no odds, therefore we can state that the population of the Universe to zero, and anyone you have ever met is merely a figment of your imagination.
We can consider the universe to be infinite from our point of reference . But we also know that the universe is mainly empty space. However, we do not know (and may never know) the true shape and extent of the universe so we cannot claim that it is truly infinite. Therefore, the mass of the universe is finite.
This question borders on theology. Based on what we know about the extent of the universe and how we may travel within it, we cannot ever know many of the answers.