One-sided L1 norm and best approximation in one-sided L1 norm

Let K be a compact subset of R, u be a alpha-finite positive measure, and f be an integrable function on K. One-sided L-1 norm is introduced, and it is defined as parallel to f parallel to = max {integral({f > 0}) vertical bar f(x)vertical bar d mu, integral({f > 0})vertical bar f(x)vertical bar d mu} A best approximation of f from, a subspace U in this norm and the uniqueness of the best approximation in this norm are discussed.