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In this paper we study the weak approximation problem of$E[\phi (x(T))] $    by$E[\phi (y(T))] $   , where$x(T)$    is the solution of a stochastic differential delay equation and$y(T)$    is defined by the Euler scheme. For$\phi \in { C}_{b}^{3} $   , Buckwar, Kuske, Mohammed and Shardlow (‘Weak convergence of the Euler scheme for stochastic differential delay equations’,  LMS J. Comput. Math.  11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence$1$   . Here we prove that the same results hold when$\phi $    is only assumed to be measurable and bounded under an additional non-degeneracy condition.

Weak approximation of stochastic differential delay equations for bounded measurable function