One often makes use of Shannon-Kolmogorov inequality in proving the consistency of the maximum likelihood estimator (MLE). The approach does not work when E(\ln f(X)) does not exist, where f is the density function of the random variable X. We consider several parametric distribution families where E(\ln f(X)) does not exist. We make use of the Kullback-Leibler (K-L) Information inequality in proving that the MLE is consistent.