Induced measure needs not to be sigma-finite

Question

Let $(\Omega, \mathcal{F}_1, \mu)$ be a $\sigma$-finite measure space, Let T:$\Omega \to \mathbb R$ be $\langle\mathcal{F}, B(R)\rangle$-measurable. Could anyone think about an example to show $\mu$'s induced measure $\mu T^{-1}$ needs not to be $\sigma$-finite?

Answer 1

Consider the case where $\mu(\Omega) = \infty$ and $T(\omega) = 0$ for all $\omega\in \Omega$. Then $\mu T^{-1} (A)$ is either $0$ or $\infty$, so we can't express $\mathbb{R}$ as a union of finite measure sets.