Shayhk, do I have to apologise for trespassing? For example, if I used someone's apartment swimming pool and I see some dropped money and I check my pocket and I see that a bill missing and I feel that it is mine, can I pick it up?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

Whoever violates the right of others must seek their pardon for that injustice; an example of this is using someone's swimming pool without their permission. He should ask for their forgiveness and pardon, as Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, "If someone has wronged a person, he has to seek his forgiveness from that injustice before the Day (of Judgment), on which there would be no dinar nor dirham. On that Day, if the unjust person has good deeds, then some will be taken from him in proportion to the injustice he did, and if he has no good deeds, then some of the bad deeds of the person he wronged will be taken from him and given to the wrongdoer to bear." [Al-Bukhaari]

As for the question of dropped money, if someone thinks that it is most likely his, then there is no harm in taking it and using it. Otherwise, he may pick it up and announce to the people that he has found that money. Then, if he finds its owner, he gives it to him, and if he does not find him after one year from making it known, then it becomes permissible for him to benefit from it.

This is in regard to something that has value and of which the heart of its owner is somewhat attached to it. But if it is an insignificant thing whose owner is not usually attached to it, then its finder may benefit from it directly without making it known, but he should guarantee it for its owner such that if its owner turns up one day, then he should give it back to him (or its value or one similar to it if available).

Please note that if a person does not repel Waswaas (obsessive whispers and doubts), they spoil his worldly life and his religion. So you should not get carried away with Waswaas and not pay any attention to it, and seek refuge with Allah from it.

Allah knows best.