Let $A_f$ be the multiplication operator in $L^2(\mathbb R)$ with the function $f$. If $g$ is a bounded Borel function on $\mathbb R$, why is $g(A_f)$ defined by the functional calculus the multiplication operator $A_{g \circ f}$ of multiplication with $g \circ f$?
For polynomials $g$ this is clear. And since polynomials are dense in the continuous functions we get it for continuous $g$. How do i get the result for characteristic and (thus again by limit argument) for bounded Borel functions? Do i need to invoke the Lusin theorem or is there another ('more elementary') argument?