For a Borel subset B of the real line, let χ χ B denote the indicator function of B . We see that the projection-valued measure μ μ X : B ( R ) → → P r ⊥ ⊥ ( L 2 ( R , C ) ) is given by μ μ X ( B ) ( ψ ψ ) = χ χ B ψ ψ , i.e., the orthogonal projection μ μ X ( B ) is the multiplication operator by the indicator function of B .