Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all finite-dimensional vector subspaces are considered negligible with respect to the Hilbert space probability measure.