Let X = ( x 1 , x 2 , … , x N ) , f : R → C X = ({x_1},{x_2}, \ldots ,{x_N}),f:{\mathbf {R}} \to {\mathbf {C}} and let P n {{\mathbf {P}}_n} be the class of polynomials of degree at most n. The generalized Christoffel function Λ n {\Lambda _n} corresponding to the measure d α d\alpha is defined by \[ Λ n ( X ; f , N , d α ) = min π ∈ P n − 1 π ( x i ) = f ( x i ) i = 1 , 2 , … , N ∫ − ∞ ∞ | π ( t ) | 2 d α ( t ) . {\Lambda _n}(X;f,N,d\alpha ) = \min \limits _{\begin {array}{*{20}{c}} {\pi \in {{\mathbf {P}}_{n - 1}}} \\ {\pi ({x_i}) = f({x_i})} \\ {i = 1,2, \ldots ,N} \\ \end {array} } \int _{ - \infty }^\infty {|\pi (t){|^2}d\alpha (t).} \] It is shown that if α \alpha satisfies some rather weak conditions then lim n → ∞ n Λ n ( X ; f , N , d α ) {\lim _{n \to \infty }}n{\Lambda _n}(X;f,N,d\alpha ) exists and the limit is also evaluated.