References a ideas behind weight functions

Question

I have some questions about weight functions:

• if we consider the $L^2$ norm with a weight of a function $f$: $$\int_\mathbb{R} w(x)|f(x)|^2 dx,$$ what is the idea of considering the weight function?
• Which properties these objects have?
• Do you have some recommendation about references for learning weight functions?

Answer 1

What you are doing, by using a weight function, is to consider $L^2$ with respect to a different measure. Roughly, if you define $$ \mu(E)=\int_{\mathbb R} 1_E(x)\,w(x)\,dx, $$ then $$ \int_{\mathbb R}|f(x)|^2\,w(x)\,dx=\int_{\mathbb R}|f(x)|^2\,d\mu. $$ Keywords are absolute continuity and Radon Nikodym Derivative.