  The Lovasz theta function provides a lower bound for the chromatic number of
finite graphs based on the solution of a semidefinite program. In this paper we
generalize it so that it gives a lower bound for the measurable chromatic
number of distance graphs on compact metric spaces.
  In particular we consider distance graphs on the unit sphere. There we
transform the original infinite semidefinite program into an infinite linear
program which then turns out to be an extremal question about Jacobi
polynomials which we solve explicitly in the limit. As an application we derive
new lower bounds for the measurable chromatic number of the Euclidean space in
dimensions 10,..., 24, and we give a new proof that it grows exponentially with
the dimension.
