  We consider the weighted average of a two dimensional CFT, whose target space
is $T^2$, over its Narain moduli space. We take as the weighing function the
integral kernel which gives rise to $T \bar T$ deformation when applied to the
world sheet moduli data of the partition function viewed as vacuum amplitude
when the world sheet is a torus. We compute the smeared partition function
where this kernel is applied to the target space moduli. Smearing the partition
function over the parameter space of a field theory generally leads to the
breakdown in the ability to write the partition function as a sum over
Boltzmann factor with unit coefficients. The weight function inspired by the $T
\bar T$ deformation appears to be an exception to this general expectation. We
show that this smearing leads to a marginal deformation corresponding to the
overall rescaling of the target space $T^2$.
