A NOTE ON A GENERAL MAXIMAL OPERATOR

Let $\mu$ be a positive Borel measure on $R^n$ which is positive on cubes. For any cube $Q \subset R^n$, a Borel measurable nonnegative function $\varphi_Q$, supported and positive a.e. with respect to $\mu$ in Q, is given. We consider a maximal function $$ M_{\mu}f(x) = sup \int \varphi Q$\mid$f$\mid$d_{\mu} $$ where the supremum is taken over all $\varphi Q$ such that $x \in Q$.