  We focus on the existence and characterization of the limit for a certain
critical branching random walks in time-space random environment in one
dimension which was introduced by M. Birnkenr et.al. Each particle performs
simple random walk on $\mathbb{Z}$ and branching mechanism depends on the
time-space site. The weak limit of this measure valued processes is
characterized as a solution to the non-trivial martingale problem and called
super-Brownian motions in random environment by L. Mytnik. Moreover, we will
show the weak uniqueness of the solutions with some initial condition.
