Under the assumption of ergodicity, the theorem of Shannon-McMillan-Breiman states that the measure theoretic entropy equal the exponential decay rate of the measure of cylinder sets (almost everywhere). This is true for every finite and generating partition. This result was subsequently generalise to infinite partitions by Carleson and Chun. In 1962 Ibragimov proved the measures of cylinder sets are lognormally distributed under the assumptions that the measure is strongly mixing and its conditional entropy function is sufficiently well approximable. We prove the CLT for the measures of cylinder sets of 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. In particular 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.