Maximal integral over observable measures

Let f be a continuous transformation on a compact, finite-dimensional manifold M, and ? a continuous function on M. This paper establishes the following formula:ess sup 1/n ?_n(x) = sup{∫?dμ|μ ∈ O_f}≤1/ness sup?_n(x),where ess sup denotes the essential supremum taken against the Lebesgue measure,?_n(x)=?(fⁱx)and O_f is the set of observable measures. Examples are provided to illustrate that the inequality could be an equality or strict. Moreover, if μ is the unique maximizing observable measure for ?, it is weakly statistical stable.