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Let      T n ( a ) = ( a i −  j ) i , j = 0 n {T_n}(a) = ({a_{i - j}})_{i,j = 0}^n       be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let      σ  n = { λ  n 0 , …  , λ  n n } {\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}       be the corresponding sets of eigenvalues of      T n ( f ) {T_n}(f)      . Define a sequence of measures      α  n , α  n ( E ) = ( n + 1 ) −  1 Σ  λ  n i ∈  E 1 , λ  n i ∈  σ  n {\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}      , and      E E       a set in the      λ  \lambda      -plane. It is shown that the weak limit      α  \alpha       of the measures      α  n {\alpha _n}       is unique and possesses at most two atoms, and the function      f f       which give rise to atoms are identified.

Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions