  The primary emphasis of this work is the development of a finite element
based space-time discretization for solving the stochastic Lagrangian averaged
Navier-Stokes (LANS-$\alpha$) equations of incompressible fluid turbulence with
multiplicative random forcing, under nonperiodic boundary conditions within a
bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The
convergence analysis of a fully discretized numerical scheme is investigated
and split into two cases according to the spacial scale $\alpha$, namely we
first assume $\alpha$ to be controlled by the step size of the space
discretization so that it vanishes when passing to the limit, then we provide
an alternative study when $\alpha$ is fixed. A preparatory analysis of uniform
estimates in both $\alpha$ and discretization parameters is carried out.
Starting out from the stochastic LANS-$\alpha$ model, we achieve convergence
toward the continuous strong solutions of the stochastic Navier-Stokes
equations in 2D when $\alpha$ vanishes at the limit. Additionally, convergence
toward the continuous strong solutions of the stochastic LANS-$\alpha$ model is
accomplished if $\alpha$ is fixed.
