Density of Uniform distribution with respect to standard-normal distribution

Question

How do I calculate the density function of the uniform distribution $U_{a,b}$ with respect to standard-normal distribution $N(0,1)$?

Answer 1

Simply divide the densities, where the density of $U_{a,b}$ is taken to be $0$ outside of $[a,b]$.

Now let me give a rigorous explanation of why this works, using basic measure theory. The Radon-Nikodym theorem says that if a distribution $\mu$ is absolutely continuous with respect to another distribution $\nu$ then there exists a function $f$ (called the density of $\mu$ with respect to $\nu$) satisfying $\mu=f\cdot \nu$. Apply this with $$\mu=\frac{1[a<x<b]}{b-a}\cdot \lambda\text{ and }\nu=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}\cdot \lambda,$$ where $\lambda$ denotes Lebesgue measure on $\mathbb R$, to conclude that $$ f=\frac{d\mu}{d\nu}=\frac{1[a<x<b]\cdot e^{-x^2/2}}{\sqrt{2\pi}(b-a)}. $$