If all of the territory of the district is included within one incorporated city, the money paid into the lighting fund shall be paid to the treasurer of that city, and administered by the legislative body thereof. The city legislative body shall have all of the powers and perform all of the duties granted to or imposed upon the board of supervisors of the county in which the district is located and shall carry out the provisions of this part as to such district to the same purpose and extent as if originally constituted the governing body thereof. The city shall be liable for all outstanding liabilities of the district incurred prior to the time the district was included within the city.