An Italian tourist was shot chasing down the thief who swiped his daughter’s necklace near the new Nets arena in Brooklyn — but cops quickly nabbed the punk, officials said yesterday.

Career criminal Tyron Lovick, 36, jumped Luciano Giuliani, 62, and his 28-year-old daughter as they walked in Fort Greene around 9:50 p.m. Thursday.

Lovick, who has 12 prior arrests, allegedly snatched her necklace and took off.

The father chased after the goon and managed to catch up with him. But the thug pulled out a .32-caliber revolver and squeezed off one round into the brave dad’s stomach, cops said.