For an ergodic system, the theorem of Shannon-McMillan-Breiman states that for every finite generating partition the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere. This was refined in 1962 by Ibragimov who proved the CLT for the measure of cylinder sets under the assumption that the measure is strong mixing and its conditional entropy function is sufficiently well approximable. In 1958 and 1960 the SMB theorem was generalised to infinite partitions. We show that the measures of cylinder sets are lognormally distributed for uniformly strong mixing systems and infinite partitions and show that the rate of convergence is polynomial. Apart from the mixing property we require that a higher than fourth moment of the information function is finite. Also, unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function. We also obtain the law of the iterated logarithm and the weak invariance principle for the information function.