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 $f$?
Yes, if f is continuous on the right. See Dunford & Schwartz V 1(1957), pp141-142. They refer to this as the Radon or Borel-Stieltjes measure determined by f.