In the classic book 'Dialogues concerning two new sciences' written in the form of a captivating conversation between the two characters Simplicio and Salviati, Galileo discusses the question of the density of air in the context of the resistance experienced by objects moving through different fluids.
Salviati says that he has found a way to determine the density of air. And it is a simple and ingenious procedure.
"I took a rather large glass bottle with a narrow neck and attached to it a leather cover, binding it tightly about the neck of the bottle. In the top of this cover I inserted and firmly fastened the valve of a leather bottle, through which I forced into the glass bottle, by means of a syringe, a large quantity of air. And since air is easily condensed one can pump into the bottle two or three times its own volume of air. After this I took an accurate balance and weighed this bottle of compressed air with the utmost precision, adjusting the weight with fine sand. I next opened the valve and allowed the compressed air to escape; then replaced the flask upon the balance and found it perceptibly lighter: from the sand which had been used as a counter- weight I now removed and laid aside as much as was necessary to again secure balance. Under these conditions there can be no doubt but that the weight of the sand thus laid aside represents the weight of the air which had been forced into the flask and had afterwards escaped."
This experiment can be easily carried out using a plastic bottle and a plastic syringe to find out the weight of air yourself, to a great degree of accuracy.