The National Football League (NFL), Major League Baseball (MLB), National Basketball Association (NBA) and the National Hockey League (NHL) are, essentially, professional sports monopolies. How have they avoided anti-trust action by the federal government? (Or have they?)
Well baseball for historical and frankly crazy decisions of SCOTUS is exempt because it is not a business that crosses state lines (by definition of the court if by no one else's).
The other sports have had their run-ins with antitrust laws; some they've won and some they've lost.
All of these have been to do with antitrust provisions in restraint of trade between leagues and clubs and clubs and players.
As for monopoly powers:
The elements of monopolization are twofold:
possession of monopoly power in a relevant market; and
willfully acquiring or maintaining that power.
There are two clear defences to allegations of running a monopoly:
the definition of the market - if the market is defined as "baseball" then there is a clear monopoly, however, if the market is "professional sports" it is not so clear that there is a monopoly. Certainly, both definitions of market are arguable.
that the monopoly was "the result of superior skill, foresight, and industry". These types of monopoly are allowed as the acquisition of monopoly power was not willfull.