Size of universe and quantity of mass

[kuartus4]

If the universe is infinite in size, is possible that it only have a finite amount of mass/energy with the rest of the universe being completely empty?

 

[nicksauce]

I can't think of a precise reason why it's impossible, but it's generally believed that the universe is the same everywhere (averaged over appropriately large scales), and there is good evidence that this is true in our observational universe.