Let $X$ be a compact metric space and write $\mathcal{B}(X)$ for the Borel $\sigma$-algebra on $X$. Also, let $(M, \mathcal{M}, \mu)$ be a measure space with $\mu(X)<\infty$. Let $\psi: M \rightarrow X$ be an $\mathcal{M}-\mathcal{B}(X)$-measurable map. Define $\mu_{\psi}: \mathcal{B}\rightarrow [0,\infty]$ by $$ \mu_{\psi} (B)= \mu (\psi^{-1}(B))$$ for $B \in \mathcal{B}(X)$.
Question Show that $\mu_{\psi}$ is well defined and that $\mu_{\psi}$ is a Borel measure on $X$.
Attempt As far as I understand I need to show that if $B\in\mathcal{B}(X)$, then $\psi^{-1}(B)\in \mathcal{M}$ in order to show that $\mu_{\psi}$ is well-defined. But I don't know how can I show that properly.
Many thanks for your help.