Let $(X, \mathcal{B}_{X}, \mu)$ be a measure space. Here, $\mu$ is **an infinite Borel measure** and **$\mu$ is not $\sigma$-finite.** Let $\pi$ be surjective Borel measurable map form $(X, \mathcal{B}_{X}, \mu)$ to a standard Borel space $(Y, \mathcal{B}_{Y})$.

My question is that whether the map: $\tau: (Y, \mathcal{B}_{Y}) \mapsto (\overline{\mathbb{R}}, \mathcal{B}_{\overline{\mathbb{R}}})$

$$y\longrightarrow \mu(\{x\in X: \pi(x)=y \})$$ is Borel measurable or not? Here $\overline{\mathbb{R}}=\mathbb{R}\cup \{+\infty\}$.

I guess the answer should be positive.

*Edit:* Thanks to the comments and I have modified $Y$ to be standard Borel.