The live trapping is a way of ensuring you DO catch them in the act though... you can't catch it in a trap if it doesn't go near the trap, so a dog in a trap is plenty of proof that the dog has been wandering onto your property.
I'm not 100% on the legalities like I said but here, a farmer is well within his rights to shoot any dog that he sees on his property. I'm pretty sure that it has to be ON his property when it's shot, as proof. Live trapping is one way of making sure of that.
A CLEAN SLATE FOR THE FUTURE