Mixture of probability measures

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

\[ P(A) = \int P_\mu(A)\, Q(\mathrm{d}\mu) \]

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

\[ P(A) = 0.7 \operatorname{normal}(0, 1)(A) + 0.3 \operatorname{normal}(1, 1)(A) \]