On Probabilities of Linear Sets

If (Xk)[infinity]k = 1 is a sequence of independent random variables with probabilities of atoms bounded away from one and E is a Borel linear subspace of [infinity], then the event {(Xk)[infinity]k = 1 [set membership, variant] E} is a.s. equivalent to the event {(Xk)nk = 1 [set membership, variant] F}, where F is an affine subspace of n for some n >= 1.