  The stochastic Auxiliary Problem Principle (APP) algorithm is a general
Stochastic Approximation (SA) scheme that turns the resolution of an original
optimization problem into the iterative resolution of a sequence of auxiliary
problems. This framework has been introduced to design
decomposition-coordination schemes but also encompasses many well-known SA
algorithms such as stochastic gradient descent or stochastic mirror descent. We
study the stochastic APP in the case where the iterates lie in a Banach space
and we consider an additive error on the computation of the subgradient of the
objective. In order to derive convergence results or efficiency estimates for a
SA scheme, the iterates must be random variables. This is why we prove the
measurability of the iterates of the stochastic APP algorithm. Then, we extend
convergence results from the Hilbert space case to the Banach space case.
Finally, we derive efficiency estimates for the function values taken at the
averaged sequence of iterates or at the last iterate, the latter being obtained
by adapting the concept of modified Fej{\'e}r monotonicity to our framework.
