A finitely additive version of the Law of the Iterated Logarithm

A finitely additive version of the law of the iterated logarithm (LIL) is proposed. The formulation involves only finite-dimensional distributions of a sequence of independent random variables (X_n)_{n\ge 1}. It is also proved that in the case where one deals with σ-additive probabilities,
the given result is equivalent to the classical version of the LIL.