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A weak value of an observable is studied for a quantum system which
is placed under the influence of an environment, where a quantum system
irreversibly evolves from a pre-selected state to a post-selected state. A
general expression for a weak value influenced by an environment is provided.
For a Markovian environment, the weak value is calculated in terms of the
predictive and retrodictive density matrices, or by means of the quantum
regression theorem. For a non-Markovian environment, a weak value is examined
by making use of exactly solvable models. It is found that although the
anomalous property is significantly suppressed by a Markovian
environment, it can survive a non-Markovian environment.