If a district contemplated by Section 11800 has not been established pursuant to Section 11500, then the term “district” as used in this chapter means the district deemed by the legislative body to be benefited by the establishment of the pedestrian mall, the boundaries of which district have been finally fixed and established in the first proceeding taken for the improvement of the pedestrian mall pursuant to the Improvement Act of 1911, the Municipal Improvement Act of 1913 or any similar special assessment law.