Now, think about picking up something underwater. It weighs less, right? It weighs less because the water it displaces is pushing up on it with a force equal to the weight of that displaced water. The same happens with the bag of air. The displaced air us pushing up on the bag, reducing its weight by the weight of the displaced air. The same volume of air is inside the bag, adding its weight to the weight of the bag. The two balance each other, so the bag of air registers the same on the scale as the bad did when it was empty.