Final answer:
Liu Bang won the final battle against Xiang Yu in 202 B.C. and declared himself the first emperor of the Han Dynasty. This event marked the beginning of a new era in Chinese history with the significant shift from aristocracy to a centralized government that recognized the support of commoners.
Step-by-step explanation:
The final battle between Liu Bang and Xiang Yu resulted in Liu Bang's victory.
After this decisive battle in 202 B.C., Liu Bang claimed his rule over the former Qin territories. Subsequently, he declared himself the first emperor of the Han Dynasty, a pivotal point in Chinese history that marked the start of a new era.
The Han Dynasty not only conquered the remaining Qin lands but also inherited its Legalist state system, which recognized the importance of commoner support for a stable government.
Liu Bang, originally a commoner himself, was instrumental in the destruction of the aristocracy and established a new social and political order, solidifying his reign by rewarding his supporters and implementing changes to create a centralized government.
Therefore, the correct answer to the question, 'What was the outcome of the final battle between Liu Bang and Xiang Yu?' is option 3: Liu Bang won the battle and declared himself the first emperor of the Han Dynasty.