Imagine if the Moon had its own moon. A moon (likely an asteroid) that lies within the Moon's Hill Sphere. I have read elsewhere about the fate of such a moon. Over time due to the Moon being tidally locked to the Earth, tidal forces would make as such that this small moon of the Moon would see its orbit shrink. This tiny object would potentially cross the Roche Limit and be ripped apart forming a ring or would simply come crashing down on the Moon.
In a scenario I am imagining, his tiny moon could have circled around our satellite for around (or more than) 100 million years before being discovered. Is it possible for this moon to survive more than 100 million years in orbit around our natural satellite before being disintegrated, or a a duration between around 4 billion years to 100 million years, assuming that this object was naturally captured by the Moon sometime in the past?