Mixture of probability measures

Let $\{P_{\mu }:\mu \in \chi \}$ be a family of probability measures and $Q$ a probability measure over $\chi$. A mixture over $\mu$ is defined such that $\mathrm{\forall }A\in \sigma (\mathrm{\Omega })$:

$$P(A)=\int P_{\mu }(A)Q(\mathrm{d}\mu )$$

For example, we can consider $\{\mathrm{normal}(0,1{)}_{\mu },\mathrm{normal}(1,1{)}_{\mu }\}$ and $\mu \sim Bernoulli(0.7)$. Then:

$$P(A)=0.7\mathrm{normal}(0,1)(A)+0.3\mathrm{normal}(1,1)(A)$$