On S. U. Bernstein's regularity type conditions in a problem of empirical Bayes approach

It is proved that the aposteriorl distribution of a random success probability $X$ in the binomial scheme can be approximated by a suitable beta-distrubution if the number $n$ of trials tends to infinity and an apriori density function of $X$ belongs to $L^r[0,1]$ for some $r\geq1$.