For finitely dominated spaces, Wall constructed a finiteness obstruction, w
hich decides whether a space is equivalent to a finite CW-complex or not. I
t was conjectured that this finiteness obstruction always vanishes for quas
i finite H-spaces, that are H-spaces whose homology looks like the homology
of a finite CW-complex. In this paper we prove this conjecture for loop sp
aces. In particular, this shows that every quasi finite loop space is actua
lly homotopy equivalent to a finite CW-complex.