Before revision:  First, we establish the inferior limit behavior of a proper lower semicontinuous L ^{0 function on a random locally convex module endowed with the locally L ^{0 topology, which makes perfect the Fenchel-Moreau duality theorem for such functions.
After revision:  First, we establish the inferior limit behavior of a proper lower semicontinuous L ^{0 function on a random locally convex module endowed with the locally L ^{0 topology, which makes perfect the Fenchel--Moreau duality theorem for such functions.