The lining, label and manufacturer tags all suggest that this is real. If you want to be extra sure, check to see if the button holes are hand sewn. Fakes rarely spend the money on hand sewn button holes.
L.P. outerwear really are superb and deserving of their premium prices. Likewise for their sport coats. Sweaters and scarves are quite good but not as great. I don't have their pants or shirts because I failed to see anything special about them.