Actually, The USA won the war with Mexico and the treaty of guadelupe Hildago took the land including Nevada and paid Mexico $15 million for it. The government ownership of the land was the basis for the homestead policy where people were given land to occupy it.
The badlands were not homesteaded. I think the existing 150 acres actually owned night be land originally obtained by homesteading.
Look at post #115. The treaty extended ‘jurisdiction’ of US boundaries, not ownership. The treaty agreed to protect property and rights of inhabitants.