A note on the scrambled set

Let (X,f) be a dynamical system, where X is a compact and metrizable space, f:X\to X is a continuous map. The following conclusions are obtained. (1) if a Borel set D is a distributional scrambled set of f and there exists an invariant probability measure \mu with \mu(D)0, then the invariant probability measure \mu is an atomic measure. (2) There exists a strongly mixing system without distributional pairs.