  This paper deals with the multivalued non-autonomous random dynamical system
generated by the non-autonomous stochastic wave equations on unbounded domains,
which has a non-Lipschitz nonlinearity with critical exponent in the three
dimensional case. We introduce the concept of weak upper semicontinuity of
multivalued functions and use such continuity to prove the measurability of
multivalued functions from a metric space to a separable Banach space. By this
approach, we show the measurability of pullback attractors of the multivalued
random dynamical system of the wave equations regardless of the completeness of
the underlying probability space. The asymptotic compactness of solutions is
proved by the method of energy equations, and the difficulty caused by the
non-compactness of Sobolev embeddings on $R^n$ is overcome by the uniform
estimates on the tails of solutions.
