I am not sure, but under Real Estate Law or Town/Property Law, if for some reason man made or mother nature, if the tree ends up falling onto your property during a storm or otherwise, then the tree becomes your problem even though it is your neigbor's tree.
So if that is the case then you should be able to pick the fruit from the part that is hanging inside your property.
I would look it up before explaining that to your neigbor if they seem to get irritated when you pick the fruit, but I am pretty sure that is the case if it is hanging over your property line then it is yours.
I hope it helps :)