After adding the air, the scale still showed the same weight. I like this one because it is one of the times when mass and weight are different at the surface of the Earth. The bag contains air, which has mass, so the combination of bag and air has more mass than just the bag by itself. Adding the air also causes the bag to take up more space, displacing some of the air around it.

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.