Final answer:
Detective Delau should obtain a search warrant based on probable cause before searching Dollree Mapp’s home, respecting the Fourth Amendment and following the precedent set by Mapp v. Ohio, which enforces the exclusionary rule.
Step-by-step explanation:
As Detective Carl Delau, suspecting Dollree Mapp's involvement in an illegal gambling business and considering the Fourth Amendment, the most appropriate action to take would be to obtain a search warrant based on probable cause before searching Dollree Mapp's home. This is in line with the legal requirements and protections established in the landmark case of Mapp v. Ohio, which determined that evidence obtained in violation of the Fourth Amendment is inadmissible in state court.
A warrantless search would not comply with constitutional requirements, unless falling under a recognized exception such as exigent circumstances or consent. Monitoring Mapp's activities could be a complementary approach but would not replace the need for a warrant to conduct a search. Seeking permission from Mapp for a voluntary search could be attempted, but it doesn't guarantee the admissibility of evidence if consent is not freely given.
The exclusionary rule, established firmly for state and federal courts in Mapp v. Ohio, ensures that illegally obtained evidence is not used in court, which underscores the importance of following proper legal protocols and respecting the rights of individuals under the suspicion of the law.