# Defining New Measure Using Monotone Function

I have a nonnegative, monotonically increasing, and bounded function $f : \mathbb{R} \to \mathbb{R}$. I want to define a measure $m'$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ (where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of $\mathbb{R}$ with the standard topology) by

$$m'(A) = m(f(A)), \:\: A \in \mathcal{B}(\mathbb{R})$$

Where $m$ is the Lebesgue measure. Can this be done for all nonnegative, monotonically increasing, and bounded functions $f$?