Simple sketch showing how, in an infinite universe, all sight lines intersect a star -- and hence the full sky should shine with the surface brightness of a star's surface.
Equivalent simple (static Euclidean) analysis for an observer in an infinite space of uniform luminosity density LV. The brightness seen at the center of a shell is independent of the distance to the shell, and hence the total brightness is infinite, since there are an infinite number of shells around us.
Figure: Whittle's (upcoming) undergraduate text.