Is it permissible to spy on the ex-wife and expose her after their divorce to prove her bad behaviors that caused him multiple problems, especially since she falsely accuses him and threatened to deport him from the country, given that the police advised him to do so as a means of defending himself?

Espionage is forbidden, unless there is a strong presumption that a crime is about to occur, in which case it is permissible to spy to prevent it, not merely to prove an accusation. If reprehensible acts are taking place in public areas such as bars, then following a woman there is not considered espionage, but rather observation of a public matter. However, if she conceals her immorality within her home, then it is not permissible to spy on her. The immorality of a wife forfeits her right to custody. Thus, there is no objection to proving a wife's immorality by photographing her or calling the police in these public places.