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.