information:classification
Step 4

Finally, connect every point on S with a half-open segment to the origin; the paradoxical decomposition of S then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center. (This center point needs a bit more care; see below.)

N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in H. However, there are only countably many such points, and like the case of the point at the center of the ball, it is possible to patch the proof to account for them all. (See below.)