Let $F: \mathbb{R} \to \mathbb{R}$ be non-decreasing, continuous, and bounded. Let $\lambda_F = \mu_F \circ F^{-1}$, where $\mu_F$ is the finite Borel measure.
- Show that $\lambda_{F | (F(- \infty), F(\infty)]} = \lambda_{|(F(- \infty), F(\infty)]}$, where $\lambda$ is the lebesgue measure.
- Let $H_x = 1_{[x, \infty)}$ and $\delta_x$ denote the Dirac measure at x. Compute $\lambda_x = \delta_x \circ H^{-1}_x$ as a Borel Measure on $\mathbb{R}$
Attempt:
I know the first part has something to do with the change of variables formula given by
- $\int_X f(g(x))d\mu(x) = \int_{\mathbb{R}} f(t)d\lambda_g(t)$, where $g:X \to \mathbb{R}$ is measurable and $f: \mathbb{R} \to [0, \infty)$ is Borel measurable