i)Let $\mu$ is a measure on $\mathbb{R}$ with the density $w$ i.e $\mu(A)=\int_{A}w(x)dx$.
Prove that, using the definition of integral that for any measurable function $f: \mathbb{R} \rightarrow \mathbb{C}$
$\int_{\mathbb{R}}f(x) d\mu(x)=\int_{\mathbb{R}}f(x) w(x) dx$.
ii) Let $F: \mathbb{R} \rightarrow (0,\infty) $ be an increasing, differentiable function, and $\mu((a,b])=F(b)-F(a)$.Prove that for any measurable function $f: \mathbb{R} \rightarrow \mathbb{C}$
$\int_{\mathbb{R}}f(x) d\mu(x)=\int_{\mathbb{R}}f(x) w(x) dx$.
I proved part i), by taking the measurable function as
Indicator function
Simple function
Non-negative function
Signed function
Complex valued function.
I prove every case except #3 Non-negative function case. Can anyone suggest some hint for this case and ii) part of the question?
