Michael Bloomberg's undercover investigation at a gun show in Arizona must be a result of the extra time the mayor has on his hands from running the well-oiled machine that is New York City, says Arizona Gov. Jan Brewer.

The governor, appearing Tuesday night on Fox News' "On the Record With Greta Van Susteren," said it's "interesting" that Bloomberg chose to launch a sting operation in the Grand Canyon state. 

"He's out here in Arizona solving our problems. I guess he believes that he has solved all the problems in New York," Brewer said.

Just weeks after the mass shooting in Tucson reopened the debate about gun control, Bloomberg used New York City funds to send undercover agents to a Phoenix gun show. The investigators captured video allegedly showing vendors failing to require background checks of gun buyers.

Federal law requires background checks on individuals before firearms may be purchased from a federally licensed dealer, manufacturer or importer. However, a "loophole" in the law allows sales at gun shows by "occasional sellers," which gun control advocates say allows anyone -- felon, drug addict, terrorist or otherwise -- to circumvent the law.

Bloomberg defended his sting by saying many of the guns purchased at gun shows in other states end up on the streets of New York City. A long-time advocate for gun control, he formed "Mayors Against Illegal Guns" in 2006, a coalition of more than 500 U.S. mayors that are dedicated to cracking down on illegal guns. 

But Arizona officials say they aren't pleased with what they consider an overstep of jurisdictional boundaries.

Maricopa County Sheriff Joe Arpaio wondered why Bloomberg didn't notify his office about any alleged violations.

"First we have elected officials sticking their noses in the immigration issue and now this," Arpaio tweeted on his Real Sheriff Joe Twitter account.

"Why is Mayor Bloomberg concerned about law enforcement in my county? Maybe I should send undercover deputies to New York," he tweeted. 

Brewer said she has no intention to send investigators to New York in response.

"I'll let him take care of his state," she said of the mayor.

She defended her state's rules on guns, which are among the most lax in the nation, saying, "Arizona's laws are sound."

Fox News' Whitney Ksiazek contributed to this report.