The police will probably correctly identify the person fleeing the scene while carrying a tv as the perpetrator. If for whatever reason the distinction is not so easy to make, they can just capture you both and ask your neighbours which one of you lives in your house. And presumably there is some way to prove that you live there, e.g. the house is on your name, there are pictures of you laying around, the dog listens to you, you know the color of the cups in the closets in the kitchen, all the clothes in the house happen to fit you... need I go on?

That assumes that a) the person flees and b) police stop to ask questions.